Properties

Label 24-637e12-1.1-c1e12-0-14
Degree $24$
Conductor $4.463\times 10^{33}$
Sign $1$
Analytic cond. $2.99915\times 10^{8}$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 6·5-s − 14·9-s + 4·13-s + 8·16-s − 4·17-s + 24·20-s − 12·23-s + 8·25-s + 4·27-s + 8·29-s + 18·31-s + 56·36-s + 42·37-s + 30·41-s + 2·43-s + 84·45-s + 42·47-s − 16·52-s + 22·53-s − 18·59-s − 28·61-s − 14·64-s − 24·65-s + 16·68-s − 24·71-s + 30·73-s + ⋯
L(s)  = 1  − 2·4-s − 2.68·5-s − 4.66·9-s + 1.10·13-s + 2·16-s − 0.970·17-s + 5.36·20-s − 2.50·23-s + 8/5·25-s + 0.769·27-s + 1.48·29-s + 3.23·31-s + 28/3·36-s + 6.90·37-s + 4.68·41-s + 0.304·43-s + 12.5·45-s + 6.12·47-s − 2.21·52-s + 3.02·53-s − 2.34·59-s − 3.58·61-s − 7/4·64-s − 2.97·65-s + 1.94·68-s − 2.84·71-s + 3.51·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{24} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(2.99915\times 10^{8}\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.524208841\)
\(L(\frac12)\) \(\approx\) \(2.524208841\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 - 4 T + 21 T^{2} - 32 T^{3} - 142 T^{4} + 924 T^{5} - 6587 T^{6} + 924 p T^{7} - 142 p^{2} T^{8} - 32 p^{3} T^{9} + 21 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
good2 \( 1 + p^{2} T^{2} + p^{3} T^{4} + 7 p T^{6} + 5 p^{2} T^{8} + 3 p^{3} T^{10} + 3 p^{3} T^{11} + 33 T^{12} + 3 p^{4} T^{13} + 3 p^{5} T^{14} + 5 p^{6} T^{16} + 7 p^{7} T^{18} + p^{11} T^{20} + p^{12} T^{22} + p^{12} T^{24} \)
3 \( ( 1 + 7 T^{2} - 2 T^{3} + 28 T^{4} + 2 T^{5} + 100 T^{6} + 2 p T^{7} + 28 p^{2} T^{8} - 2 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
5 \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 259 T^{4} + 576 T^{5} + 958 T^{6} + 876 T^{7} - 224 p T^{8} - 342 p^{2} T^{9} - 27077 T^{10} - 70008 T^{11} - 159154 T^{12} - 70008 p T^{13} - 27077 p^{2} T^{14} - 342 p^{5} T^{15} - 224 p^{5} T^{16} + 876 p^{5} T^{17} + 958 p^{6} T^{18} + 576 p^{7} T^{19} + 259 p^{8} T^{20} + 96 p^{9} T^{21} + 28 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 82 T^{2} + 3073 T^{4} - 69688 T^{6} + 1086920 T^{8} - 1196198 p T^{10} + 144840644 T^{12} - 1196198 p^{3} T^{14} + 1086920 p^{4} T^{16} - 69688 p^{6} T^{18} + 3073 p^{8} T^{20} - 82 p^{10} T^{22} + p^{12} T^{24} \)
17 \( 1 + 4 T - 65 T^{2} - 236 T^{3} + 2367 T^{4} + 6812 T^{5} - 71424 T^{6} - 136128 T^{7} + 1875161 T^{8} + 2094308 T^{9} - 39826735 T^{10} - 15268952 T^{11} + 716620078 T^{12} - 15268952 p T^{13} - 39826735 p^{2} T^{14} + 2094308 p^{3} T^{15} + 1875161 p^{4} T^{16} - 136128 p^{5} T^{17} - 71424 p^{6} T^{18} + 6812 p^{7} T^{19} + 2367 p^{8} T^{20} - 236 p^{9} T^{21} - 65 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 - 170 T^{2} + 13833 T^{4} - 714484 T^{6} + 26229656 T^{8} - 725388054 T^{10} + 15570602740 T^{12} - 725388054 p^{2} T^{14} + 26229656 p^{4} T^{16} - 714484 p^{6} T^{18} + 13833 p^{8} T^{20} - 170 p^{10} T^{22} + p^{12} T^{24} \)
23 \( 1 + 12 T + 26 T^{2} - 128 T^{3} + 181 T^{4} + 6600 T^{5} + 14926 T^{6} - 100428 T^{7} - 552326 T^{8} + 2571636 T^{9} + 33604322 T^{10} + 2318056 p T^{11} - 320055507 T^{12} + 2318056 p^{2} T^{13} + 33604322 p^{2} T^{14} + 2571636 p^{3} T^{15} - 552326 p^{4} T^{16} - 100428 p^{5} T^{17} + 14926 p^{6} T^{18} + 6600 p^{7} T^{19} + 181 p^{8} T^{20} - 128 p^{9} T^{21} + 26 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 8 T - 66 T^{2} + 148 T^{3} + 6049 T^{4} + 1990 T^{5} - 212968 T^{6} - 874644 T^{7} + 5366152 T^{8} + 28797506 T^{9} + 6167763 T^{10} - 559858476 T^{11} - 959497590 T^{12} - 559858476 p T^{13} + 6167763 p^{2} T^{14} + 28797506 p^{3} T^{15} + 5366152 p^{4} T^{16} - 874644 p^{5} T^{17} - 212968 p^{6} T^{18} + 1990 p^{7} T^{19} + 6049 p^{8} T^{20} + 148 p^{9} T^{21} - 66 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 18 T + 280 T^{2} - 3096 T^{3} + 30816 T^{4} - 265842 T^{5} + 2121848 T^{6} - 15604506 T^{7} + 107979056 T^{8} - 706238856 T^{9} + 4401177000 T^{10} - 26180493402 T^{11} + 149120774974 T^{12} - 26180493402 p T^{13} + 4401177000 p^{2} T^{14} - 706238856 p^{3} T^{15} + 107979056 p^{4} T^{16} - 15604506 p^{5} T^{17} + 2121848 p^{6} T^{18} - 265842 p^{7} T^{19} + 30816 p^{8} T^{20} - 3096 p^{9} T^{21} + 280 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 42 T + 945 T^{2} - 14994 T^{3} + 185598 T^{4} - 51222 p T^{5} + 16605131 T^{6} - 128997846 T^{7} + 917274780 T^{8} - 6151787586 T^{9} + 39790847349 T^{10} - 250948684746 T^{11} + 1545474799320 T^{12} - 250948684746 p T^{13} + 39790847349 p^{2} T^{14} - 6151787586 p^{3} T^{15} + 917274780 p^{4} T^{16} - 128997846 p^{5} T^{17} + 16605131 p^{6} T^{18} - 51222 p^{8} T^{19} + 185598 p^{8} T^{20} - 14994 p^{9} T^{21} + 945 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 - 30 T + 561 T^{2} - 7830 T^{3} + 90070 T^{4} - 887970 T^{5} + 7840739 T^{6} - 63891690 T^{7} + 493300124 T^{8} - 3663998070 T^{9} + 26303779525 T^{10} - 181274688750 T^{11} + 1190670941216 T^{12} - 181274688750 p T^{13} + 26303779525 p^{2} T^{14} - 3663998070 p^{3} T^{15} + 493300124 p^{4} T^{16} - 63891690 p^{5} T^{17} + 7840739 p^{6} T^{18} - 887970 p^{7} T^{19} + 90070 p^{8} T^{20} - 7830 p^{9} T^{21} + 561 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 2 T - 145 T^{2} + 382 T^{3} + 10113 T^{4} - 26622 T^{5} - 446076 T^{6} + 446446 T^{7} + 17414132 T^{8} + 28824226 T^{9} - 865196753 T^{10} - 1024914406 T^{11} + 42721342945 T^{12} - 1024914406 p T^{13} - 865196753 p^{2} T^{14} + 28824226 p^{3} T^{15} + 17414132 p^{4} T^{16} + 446446 p^{5} T^{17} - 446076 p^{6} T^{18} - 26622 p^{7} T^{19} + 10113 p^{8} T^{20} + 382 p^{9} T^{21} - 145 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 - 42 T + 1028 T^{2} - 18480 T^{3} + 5756 p T^{4} - 3388170 T^{5} + 37482200 T^{6} - 374122674 T^{7} + 3422821532 T^{8} - 29033886912 T^{9} + 230318691500 T^{10} - 1719234646386 T^{11} + 12120532870718 T^{12} - 1719234646386 p T^{13} + 230318691500 p^{2} T^{14} - 29033886912 p^{3} T^{15} + 3422821532 p^{4} T^{16} - 374122674 p^{5} T^{17} + 37482200 p^{6} T^{18} - 3388170 p^{7} T^{19} + 5756 p^{9} T^{20} - 18480 p^{9} T^{21} + 1028 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 22 T + 75 T^{2} + 1262 T^{3} - 3653 T^{4} - 56896 T^{5} - 435976 T^{6} + 3899676 T^{7} + 72557701 T^{8} - 437078294 T^{9} - 3368488995 T^{10} + 12741643314 T^{11} + 134482979838 T^{12} + 12741643314 p T^{13} - 3368488995 p^{2} T^{14} - 437078294 p^{3} T^{15} + 72557701 p^{4} T^{16} + 3899676 p^{5} T^{17} - 435976 p^{6} T^{18} - 56896 p^{7} T^{19} - 3653 p^{8} T^{20} + 1262 p^{9} T^{21} + 75 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 18 T + 352 T^{2} + 4392 T^{3} + 50804 T^{4} + 462762 T^{5} + 4098272 T^{6} + 31242690 T^{7} + 257004644 T^{8} + 2067787080 T^{9} + 18153739368 T^{10} + 149323686858 T^{11} + 1224311625150 T^{12} + 149323686858 p T^{13} + 18153739368 p^{2} T^{14} + 2067787080 p^{3} T^{15} + 257004644 p^{4} T^{16} + 31242690 p^{5} T^{17} + 4098272 p^{6} T^{18} + 462762 p^{7} T^{19} + 50804 p^{8} T^{20} + 4392 p^{9} T^{21} + 352 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
61 \( ( 1 + 14 T + 279 T^{2} + 2854 T^{3} + 32699 T^{4} + 263412 T^{5} + 2369290 T^{6} + 263412 p T^{7} + 32699 p^{2} T^{8} + 2854 p^{3} T^{9} + 279 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 - 416 T^{2} + 89024 T^{4} - 12805108 T^{6} + 1386054656 T^{8} - 120600192288 T^{10} + 8774166644694 T^{12} - 120600192288 p^{2} T^{14} + 1386054656 p^{4} T^{16} - 12805108 p^{6} T^{18} + 89024 p^{8} T^{20} - 416 p^{10} T^{22} + p^{12} T^{24} \)
71 \( 1 + 24 T + 638 T^{2} + 10704 T^{3} + 181573 T^{4} + 2429664 T^{5} + 32494394 T^{6} + 370328904 T^{7} + 4209378170 T^{8} + 42185547816 T^{9} + 421044910550 T^{10} + 3761130662208 T^{11} + 33433168087229 T^{12} + 3761130662208 p T^{13} + 421044910550 p^{2} T^{14} + 42185547816 p^{3} T^{15} + 4209378170 p^{4} T^{16} + 370328904 p^{5} T^{17} + 32494394 p^{6} T^{18} + 2429664 p^{7} T^{19} + 181573 p^{8} T^{20} + 10704 p^{9} T^{21} + 638 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 30 T + 721 T^{2} - 12630 T^{3} + 191678 T^{4} - 2496834 T^{5} + 29784443 T^{6} - 324024042 T^{7} + 3341072420 T^{8} - 32480342070 T^{9} + 304534332021 T^{10} - 2733970220238 T^{11} + 23825878427568 T^{12} - 2733970220238 p T^{13} + 304534332021 p^{2} T^{14} - 32480342070 p^{3} T^{15} + 3341072420 p^{4} T^{16} - 324024042 p^{5} T^{17} + 29784443 p^{6} T^{18} - 2496834 p^{7} T^{19} + 191678 p^{8} T^{20} - 12630 p^{9} T^{21} + 721 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 28 T + 98 T^{2} + 3296 T^{3} - 3483 T^{4} - 608232 T^{5} + 2364678 T^{6} + 39793724 T^{7} + 3691514 T^{8} - 4430539972 T^{9} + 14465689162 T^{10} + 35457502408 T^{11} + 275896569661 T^{12} + 35457502408 p T^{13} + 14465689162 p^{2} T^{14} - 4430539972 p^{3} T^{15} + 3691514 p^{4} T^{16} + 39793724 p^{5} T^{17} + 2364678 p^{6} T^{18} - 608232 p^{7} T^{19} - 3483 p^{8} T^{20} + 3296 p^{9} T^{21} + 98 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 692 T^{2} + 237440 T^{4} - 52943716 T^{6} + 8500384712 T^{8} - 1032128260020 T^{10} + 97020933033606 T^{12} - 1032128260020 p^{2} T^{14} + 8500384712 p^{4} T^{16} - 52943716 p^{6} T^{18} + 237440 p^{8} T^{20} - 692 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 - 12 T + 277 T^{2} - 2748 T^{3} + 36829 T^{4} - 397104 T^{5} + 2770552 T^{6} - 23558250 T^{7} - 8773336 T^{8} + 898812894 T^{9} - 22978757675 T^{10} + 355473536682 T^{11} - 3230803249795 T^{12} + 355473536682 p T^{13} - 22978757675 p^{2} T^{14} + 898812894 p^{3} T^{15} - 8773336 p^{4} T^{16} - 23558250 p^{5} T^{17} + 2770552 p^{6} T^{18} - 397104 p^{7} T^{19} + 36829 p^{8} T^{20} - 2748 p^{9} T^{21} + 277 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 6 T + 409 T^{2} - 2382 T^{3} + 87881 T^{4} - 349674 T^{5} + 12207200 T^{6} - 13708968 T^{7} + 1176348524 T^{8} + 4020177528 T^{9} + 86854874625 T^{10} + 857401403916 T^{11} + 6767014810365 T^{12} + 857401403916 p T^{13} + 86854874625 p^{2} T^{14} + 4020177528 p^{3} T^{15} + 1176348524 p^{4} T^{16} - 13708968 p^{5} T^{17} + 12207200 p^{6} T^{18} - 349674 p^{7} T^{19} + 87881 p^{8} T^{20} - 2382 p^{9} T^{21} + 409 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.63311802568387077990621226236, −3.54418810326667881017476104109, −3.33491408374266540928933268194, −2.93871175384895352020803147107, −2.90779598767480293027265940775, −2.87301618836448227733386951655, −2.84271668321121524128786644910, −2.80917700085769624832829420419, −2.63222759022414081679180807034, −2.62800958468126420482350938363, −2.61548574411380848360363859281, −2.45673927399404249464186668486, −2.34907576689520637293542276034, −2.07597237543405583232259579481, −2.01374460532462980798950292803, −1.82760080674323987089260965756, −1.78919787609118524732908592654, −1.32305892894279036268625861411, −1.10101692962524911214673654566, −0.891702457905492016387122306263, −0.66292751111619430780888216971, −0.62997777878567892157913208409, −0.62485625325438715576426232441, −0.59391530321083812751228959264, −0.43665822009123183739497436232, 0.43665822009123183739497436232, 0.59391530321083812751228959264, 0.62485625325438715576426232441, 0.62997777878567892157913208409, 0.66292751111619430780888216971, 0.891702457905492016387122306263, 1.10101692962524911214673654566, 1.32305892894279036268625861411, 1.78919787609118524732908592654, 1.82760080674323987089260965756, 2.01374460532462980798950292803, 2.07597237543405583232259579481, 2.34907576689520637293542276034, 2.45673927399404249464186668486, 2.61548574411380848360363859281, 2.62800958468126420482350938363, 2.63222759022414081679180807034, 2.80917700085769624832829420419, 2.84271668321121524128786644910, 2.87301618836448227733386951655, 2.90779598767480293027265940775, 2.93871175384895352020803147107, 3.33491408374266540928933268194, 3.54418810326667881017476104109, 3.63311802568387077990621226236

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.