Properties

Label 32-288e16-1.1-c1e16-0-1
Degree $32$
Conductor $2.240\times 10^{39}$
Sign $1$
Analytic cond. $611946.$
Root an. cond. $1.51647$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 15·9-s − 12·11-s + 4·19-s + 13·25-s + 30·27-s − 72·33-s − 36·41-s − 8·43-s − 23·49-s + 24·57-s − 12·59-s + 16·67-s − 4·73-s + 78·75-s + 75·81-s − 54·83-s + 8·97-s − 180·99-s − 6·113-s + 4·121-s − 216·123-s + 127-s − 48·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 3.46·3-s + 5·9-s − 3.61·11-s + 0.917·19-s + 13/5·25-s + 5.77·27-s − 12.5·33-s − 5.62·41-s − 1.21·43-s − 3.28·49-s + 3.17·57-s − 1.56·59-s + 1.95·67-s − 0.468·73-s + 9.00·75-s + 25/3·81-s − 5.92·83-s + 0.812·97-s − 18.0·99-s − 0.564·113-s + 4/11·121-s − 19.4·123-s + 0.0887·127-s − 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{80} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(611946.\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{80} \cdot 3^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.351900682\)
\(L(\frac12)\) \(\approx\) \(4.351900682\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - p T + 2 p T^{2} - 5 p T^{3} + 10 p T^{4} - 5 p^{2} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \)
good5 \( 1 - 13 T^{2} + 48 T^{4} + 103 T^{6} - 1099 T^{8} + 3648 T^{10} - 5222 T^{12} - 177298 T^{14} + 1667616 T^{16} - 177298 p^{2} T^{18} - 5222 p^{4} T^{20} + 3648 p^{6} T^{22} - 1099 p^{8} T^{24} + 103 p^{10} T^{26} + 48 p^{12} T^{28} - 13 p^{14} T^{30} + p^{16} T^{32} \)
7 \( 1 + 23 T^{2} + 204 T^{4} + 1399 T^{6} + 12029 T^{8} + 45000 T^{10} - 343382 T^{12} - 82438 p^{2} T^{14} - 24407256 T^{16} - 82438 p^{4} T^{18} - 343382 p^{4} T^{20} + 45000 p^{6} T^{22} + 12029 p^{8} T^{24} + 1399 p^{10} T^{26} + 204 p^{12} T^{28} + 23 p^{14} T^{30} + p^{16} T^{32} \)
11 \( ( 1 - 3 T + 38 T^{2} - 87 T^{3} + 606 T^{4} - 87 p T^{5} + 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 9 T + 41 T^{2} + 108 T^{3} + 276 T^{4} + 108 p T^{5} + 41 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 47 T^{2} + 1116 T^{4} + 17239 T^{6} + 168353 T^{8} + 463056 T^{10} - 2070506 p T^{12} - 784598686 T^{14} - 12569281176 T^{16} - 784598686 p^{2} T^{18} - 2070506 p^{5} T^{20} + 463056 p^{6} T^{22} + 168353 p^{8} T^{24} + 17239 p^{10} T^{26} + 1116 p^{12} T^{28} + 47 p^{14} T^{30} + p^{16} T^{32} \)
17 \( ( 1 - 101 T^{2} + 4882 T^{4} - 146891 T^{6} + 2998666 T^{8} - 146891 p^{2} T^{10} + 4882 p^{4} T^{12} - 101 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - T + 64 T^{2} - 49 T^{3} + 1726 T^{4} - 49 p T^{5} + 64 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \)
23 \( 1 - 85 T^{2} + 3432 T^{4} - 80441 T^{6} + 1017209 T^{8} + 1764696 T^{10} - 386280290 T^{12} + 9323832998 T^{14} - 183562178736 T^{16} + 9323832998 p^{2} T^{18} - 386280290 p^{4} T^{20} + 1764696 p^{6} T^{22} + 1017209 p^{8} T^{24} - 80441 p^{10} T^{26} + 3432 p^{12} T^{28} - 85 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 97 T^{2} + 3780 T^{4} - 80561 T^{6} + 1632089 T^{8} - 54786000 T^{10} + 1038848902 T^{12} + 18974156858 T^{14} - 1329857348520 T^{16} + 18974156858 p^{2} T^{18} + 1038848902 p^{4} T^{20} - 54786000 p^{6} T^{22} + 1632089 p^{8} T^{24} - 80561 p^{10} T^{26} + 3780 p^{12} T^{28} - 97 p^{14} T^{30} + p^{16} T^{32} \)
31 \( 1 + 131 T^{2} + 7152 T^{4} + 312151 T^{6} + 16524593 T^{8} + 706250232 T^{10} + 22761450214 T^{12} + 818587141454 T^{14} + 29243993208000 T^{16} + 818587141454 p^{2} T^{18} + 22761450214 p^{4} T^{20} + 706250232 p^{6} T^{22} + 16524593 p^{8} T^{24} + 312151 p^{10} T^{26} + 7152 p^{12} T^{28} + 131 p^{14} T^{30} + p^{16} T^{32} \)
37 \( ( 1 - 140 T^{2} + 8596 T^{4} - 317540 T^{6} + 10470934 T^{8} - 317540 p^{2} T^{10} + 8596 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 18 T + 292 T^{2} + 3312 T^{3} + 34963 T^{4} + 303732 T^{5} + 2503888 T^{6} + 17908938 T^{7} + 122450608 T^{8} + 17908938 p T^{9} + 2503888 p^{2} T^{10} + 303732 p^{3} T^{11} + 34963 p^{4} T^{12} + 3312 p^{5} T^{13} + 292 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 4 T - 96 T^{2} - 868 T^{3} + 4061 T^{4} + 55182 T^{5} + 78652 T^{6} - 1341518 T^{7} - 8451684 T^{8} - 1341518 p T^{9} + 78652 p^{2} T^{10} + 55182 p^{3} T^{11} + 4061 p^{4} T^{12} - 868 p^{5} T^{13} - 96 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 - 265 T^{2} + 38172 T^{4} - 3663305 T^{6} + 256619789 T^{8} - 13483362840 T^{10} + 543517455226 T^{12} - 17957421673750 T^{14} + 672442507889160 T^{16} - 17957421673750 p^{2} T^{18} + 543517455226 p^{4} T^{20} - 13483362840 p^{6} T^{22} + 256619789 p^{8} T^{24} - 3663305 p^{10} T^{26} + 38172 p^{12} T^{28} - 265 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 + 196 T^{2} + 21556 T^{4} + 1665484 T^{6} + 98315734 T^{8} + 1665484 p^{2} T^{10} + 21556 p^{4} T^{12} + 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 6 T + 178 T^{2} + 996 T^{3} + 269 p T^{4} + 89238 T^{5} + 1194346 T^{6} + 6437052 T^{7} + 79012984 T^{8} + 6437052 p T^{9} + 1194346 p^{2} T^{10} + 89238 p^{3} T^{11} + 269 p^{5} T^{12} + 996 p^{5} T^{13} + 178 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( 1 + 299 T^{2} + 43416 T^{4} + 4465735 T^{6} + 393470093 T^{8} + 31461221184 T^{10} + 2332722797890 T^{12} + 164022774523334 T^{14} + 10603540284680400 T^{16} + 164022774523334 p^{2} T^{18} + 2332722797890 p^{4} T^{20} + 31461221184 p^{6} T^{22} + 393470093 p^{8} T^{24} + 4465735 p^{10} T^{26} + 43416 p^{12} T^{28} + 299 p^{14} T^{30} + p^{16} T^{32} \)
67 \( ( 1 - 8 T - 138 T^{2} + 1052 T^{3} + 11279 T^{4} - 57198 T^{5} - 930218 T^{6} + 1298482 T^{7} + 71382744 T^{8} + 1298482 p T^{9} - 930218 p^{2} T^{10} - 57198 p^{3} T^{11} + 11279 p^{4} T^{12} + 1052 p^{5} T^{13} - 138 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 400 T^{2} + 73324 T^{4} + 8374048 T^{6} + 685441990 T^{8} + 8374048 p^{2} T^{10} + 73324 p^{4} T^{12} + 400 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 + T + 214 T^{2} - 5 T^{3} + 20758 T^{4} - 5 p T^{5} + 214 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( 1 + 383 T^{2} + 74724 T^{4} + 9593503 T^{6} + 916548293 T^{8} + 73505234952 T^{10} + 5745220732498 T^{12} + 475438917658202 T^{14} + 38877973133064792 T^{16} + 475438917658202 p^{2} T^{18} + 5745220732498 p^{4} T^{20} + 73505234952 p^{6} T^{22} + 916548293 p^{8} T^{24} + 9593503 p^{10} T^{26} + 74724 p^{12} T^{28} + 383 p^{14} T^{30} + p^{16} T^{32} \)
83 \( ( 1 + 27 T + 544 T^{2} + 8127 T^{3} + 107593 T^{4} + 1304076 T^{5} + 14403022 T^{6} + 148987620 T^{7} + 1398163588 T^{8} + 148987620 p T^{9} + 14403022 p^{2} T^{10} + 1304076 p^{3} T^{11} + 107593 p^{4} T^{12} + 8127 p^{5} T^{13} + 544 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 440 T^{2} + 100348 T^{4} - 14946776 T^{6} + 1566592486 T^{8} - 14946776 p^{2} T^{10} + 100348 p^{4} T^{12} - 440 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 4 T - 198 T^{2} - 320 T^{3} + 21017 T^{4} + 106308 T^{5} - 1078406 T^{6} - 6931984 T^{7} + 64836612 T^{8} - 6931984 p T^{9} - 1078406 p^{2} T^{10} + 106308 p^{3} T^{11} + 21017 p^{4} T^{12} - 320 p^{5} T^{13} - 198 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.29669173497405712967341415373, −3.18632899413497466182785682679, −3.12305752427886770923808050774, −3.01346848289248341381324917959, −2.98910087280652977677542355886, −2.95436559968820502108632481714, −2.88943942536569510494639178436, −2.81331566885800559844710971885, −2.70497940532863656314890070475, −2.47457251657554182071244831037, −2.38835484256368668636953070769, −2.38611835807391051525121408427, −2.34123238846219232447366373510, −2.28371379343015997617721740178, −2.03861373431914443137651071921, −1.82455123387258135573811726061, −1.80925375524667014239409419624, −1.57318145223686444026515278702, −1.50373819484859003868790503317, −1.47077873464823914621433879403, −1.41158412030786931712559856285, −1.08019953706045669843823091059, −0.922414244749295670789846751327, −0.33117559347902012080451403927, −0.30871919388493928437035734948, 0.30871919388493928437035734948, 0.33117559347902012080451403927, 0.922414244749295670789846751327, 1.08019953706045669843823091059, 1.41158412030786931712559856285, 1.47077873464823914621433879403, 1.50373819484859003868790503317, 1.57318145223686444026515278702, 1.80925375524667014239409419624, 1.82455123387258135573811726061, 2.03861373431914443137651071921, 2.28371379343015997617721740178, 2.34123238846219232447366373510, 2.38611835807391051525121408427, 2.38835484256368668636953070769, 2.47457251657554182071244831037, 2.70497940532863656314890070475, 2.81331566885800559844710971885, 2.88943942536569510494639178436, 2.95436559968820502108632481714, 2.98910087280652977677542355886, 3.01346848289248341381324917959, 3.12305752427886770923808050774, 3.18632899413497466182785682679, 3.29669173497405712967341415373

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

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