Properties

Label 24-637e12-1.1-c1e12-0-7
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  + 8·4-s + 6·5-s + 7·9-s − 6·11-s + 4·13-s + 32·16-s + 8·17-s + 48·20-s + 24·23-s + 8·25-s + 4·27-s + 8·29-s − 18·31-s + 56·36-s + 30·41-s + 2·43-s − 48·44-s + 42·45-s − 42·47-s + 32·52-s + 22·53-s − 36·55-s + 14·61-s + 82·64-s + 24·65-s + 24·67-s + 64·68-s + ⋯
L(s)  = 1  + 4·4-s + 2.68·5-s + 7/3·9-s − 1.80·11-s + 1.10·13-s + 8·16-s + 1.94·17-s + 10.7·20-s + 5.00·23-s + 8/5·25-s + 0.769·27-s + 1.48·29-s − 3.23·31-s + 28/3·36-s + 4.68·41-s + 0.304·43-s − 7.23·44-s + 6.26·45-s − 6.12·47-s + 4.43·52-s + 3.02·53-s − 4.85·55-s + 1.79·61-s + 41/4·64-s + 2.97·65-s + 2.93·67-s + 7.76·68-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

\(L(1)\) \(\approx\) \(40.38734147\)
\(L(\frac12)\) \(\approx\) \(40.38734147\)
\(L(\frac{3}{2})\) 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 \( 1 - p^{3} T^{2} + p^{5} T^{4} - 41 p T^{6} + 19 p^{3} T^{8} - 15 p^{4} T^{10} + 417 T^{12} - 15 p^{6} T^{14} + 19 p^{7} T^{16} - 41 p^{7} T^{18} + p^{13} T^{20} - p^{13} T^{22} + p^{12} T^{24} \)
3 \( 1 - 7 T^{2} - 4 T^{3} + 7 p T^{4} + 4 p T^{5} + 8 T^{6} + 26 p T^{7} - 164 T^{8} - 202 p T^{9} + 59 p T^{10} + 1018 T^{11} + 637 T^{12} + 1018 p T^{13} + 59 p^{3} T^{14} - 202 p^{4} T^{15} - 164 p^{4} T^{16} + 26 p^{6} T^{17} + 8 p^{6} T^{18} + 4 p^{8} T^{19} + 7 p^{9} T^{20} - 4 p^{9} T^{21} - 7 p^{10} T^{22} + p^{12} T^{24} \)
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 \( 1 + 6 T + 59 T^{2} + 282 T^{3} + 1777 T^{4} + 7782 T^{5} + 37916 T^{6} + 156246 T^{7} + 637088 T^{8} + 2504538 T^{9} + 8930699 T^{10} + 33037974 T^{11} + 106097225 T^{12} + 33037974 p T^{13} + 8930699 p^{2} T^{14} + 2504538 p^{3} T^{15} + 637088 p^{4} T^{16} + 156246 p^{5} T^{17} + 37916 p^{6} T^{18} + 7782 p^{7} T^{19} + 1777 p^{8} T^{20} + 282 p^{9} T^{21} + 59 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
17 \( ( 1 - 4 T + 81 T^{2} - 280 T^{3} + 3074 T^{4} - 8724 T^{5} + 67033 T^{6} - 8724 p T^{7} + 3074 p^{2} T^{8} - 280 p^{3} T^{9} + 81 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
19 \( 1 + 85 T^{2} + 3921 T^{4} - 3444 T^{5} + 128036 T^{6} - 237714 T^{7} + 3259796 T^{8} - 8826714 T^{9} + 72199845 T^{10} - 227211318 T^{11} + 1442326981 T^{12} - 227211318 p T^{13} + 72199845 p^{2} T^{14} - 8826714 p^{3} T^{15} + 3259796 p^{4} T^{16} - 237714 p^{5} T^{17} + 128036 p^{6} T^{18} - 3444 p^{7} T^{19} + 3921 p^{8} T^{20} + 85 p^{10} T^{22} + p^{12} T^{24} \)
23 \( ( 1 - 12 T + 118 T^{2} - 772 T^{3} + 4479 T^{4} - 1000 p T^{5} + 111732 T^{6} - 1000 p^{2} T^{7} + 4479 p^{2} T^{8} - 772 p^{3} T^{9} + 118 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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 \( 1 - 126 T^{2} + 11151 T^{4} - 716278 T^{6} + 38725515 T^{8} - 1787180508 T^{10} + 71085787530 T^{12} - 1787180508 p^{2} T^{14} + 38725515 p^{4} T^{16} - 716278 p^{6} T^{18} + 11151 p^{8} T^{20} - 126 p^{10} T^{22} + p^{12} T^{24} \)
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 \( 1 - 380 T^{2} + 77000 T^{4} - 10623268 T^{6} + 1101557216 T^{8} - 89823025164 T^{10} + 5891147678118 T^{12} - 89823025164 p^{2} T^{14} + 1101557216 p^{4} T^{16} - 10623268 p^{6} T^{18} + 77000 p^{8} T^{20} - 380 p^{10} T^{22} + p^{12} T^{24} \)
61 \( 1 - 14 T - 83 T^{2} + 1802 T^{3} + 5186 T^{4} - 144106 T^{5} + 73107 T^{6} + 4422758 T^{7} - 15350440 T^{8} + 324186 p T^{9} + 1015299113 T^{10} - 4928294278 T^{11} - 40198716168 T^{12} - 4928294278 p T^{13} + 1015299113 p^{2} T^{14} + 324186 p^{4} T^{15} - 15350440 p^{4} T^{16} + 4422758 p^{5} T^{17} + 73107 p^{6} T^{18} - 144106 p^{7} T^{19} + 5186 p^{8} T^{20} + 1802 p^{9} T^{21} - 83 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 - 24 T + 496 T^{2} - 7296 T^{3} + 94112 T^{4} - 1067976 T^{5} + 11279564 T^{6} - 113091816 T^{7} + 1089104960 T^{8} - 10093825152 T^{9} + 89600000688 T^{10} - 772079923320 T^{11} + 6378740648022 T^{12} - 772079923320 p T^{13} + 89600000688 p^{2} T^{14} - 10093825152 p^{3} T^{15} + 1089104960 p^{4} T^{16} - 113091816 p^{5} T^{17} + 11279564 p^{6} T^{18} - 1067976 p^{7} T^{19} + 94112 p^{8} T^{20} - 7296 p^{9} T^{21} + 496 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \)
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 \( 1 - 410 T^{2} + 83665 T^{4} - 12720140 T^{6} + 1658742032 T^{8} - 183597109502 T^{10} + 17425244531684 T^{12} - 183597109502 p^{2} T^{14} + 1658742032 p^{4} T^{16} - 12720140 p^{6} T^{18} + 83665 p^{8} T^{20} - 410 p^{10} T^{22} + p^{12} T^{24} \)
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

−3.44082843879742382614576330610, −3.37293002575133308890356312071, −3.27454191755249299511368993266, −3.16005083753439464613862458541, −2.89910144095185311192409752383, −2.76268432667033013246204254220, −2.65184053021041713941785789926, −2.62558530090715252721349774550, −2.58049170946109511641228966642, −2.48947337808136539436399652491, −2.42198744320962816878833523516, −2.26044596915791086552431472692, −2.21784842121203748299058119153, −2.18555782872539672204316516091, −1.96575490099697279108910162980, −1.90199270524595445907021938291, −1.52048570100538537448015918617, −1.49106319130641087871097005855, −1.26382935065783460890187967880, −1.26170640179865343641330087164, −1.22523374295791013194966675798, −1.16892561846966301288255666567, −1.13272787588877137482176456384, −0.875022971533672723103547542580, −0.18629716494698232314646706253, 0.18629716494698232314646706253, 0.875022971533672723103547542580, 1.13272787588877137482176456384, 1.16892561846966301288255666567, 1.22523374295791013194966675798, 1.26170640179865343641330087164, 1.26382935065783460890187967880, 1.49106319130641087871097005855, 1.52048570100538537448015918617, 1.90199270524595445907021938291, 1.96575490099697279108910162980, 2.18555782872539672204316516091, 2.21784842121203748299058119153, 2.26044596915791086552431472692, 2.42198744320962816878833523516, 2.48947337808136539436399652491, 2.58049170946109511641228966642, 2.62558530090715252721349774550, 2.65184053021041713941785789926, 2.76268432667033013246204254220, 2.89910144095185311192409752383, 3.16005083753439464613862458541, 3.27454191755249299511368993266, 3.37293002575133308890356312071, 3.44082843879742382614576330610

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

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