Properties

Label 32-42e32-1.1-c3e16-0-1
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $1.89598\times 10^{32}$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 788·25-s − 152·37-s + 1.40e3·43-s + 3.05e3·67-s + 728·79-s − 1.54e3·109-s + 1.16e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.95e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 6.30·25-s − 0.675·37-s + 4.99·43-s + 5.57·67-s + 1.03·79-s − 1.35·109-s + 8.77·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 13.4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(1.89598\times 10^{32}\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(194.5231356\)
\(L(\frac12)\) \(\approx\) \(194.5231356\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + 394 T^{2} + 75853 T^{4} + 1675022 p T^{6} + 880531096 T^{8} + 1675022 p^{7} T^{10} + 75853 p^{12} T^{12} + 394 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
11 \( ( 1 - 5842 T^{2} + 16417117 T^{4} - 30523502158 T^{6} + 44443300773688 T^{8} - 30523502158 p^{6} T^{10} + 16417117 p^{12} T^{12} - 5842 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
13 \( ( 1 - 14750 T^{2} + 100165933 T^{4} - 31441942490 p T^{6} + 1095322120237240 T^{8} - 31441942490 p^{7} T^{10} + 100165933 p^{12} T^{12} - 14750 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
17 \( ( 1 + 17908 T^{2} + 136974664 T^{4} + 557552341852 T^{6} + 2018744999711374 T^{8} + 557552341852 p^{6} T^{10} + 136974664 p^{12} T^{12} + 17908 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
19 \( ( 1 - 24014 T^{2} + 309658861 T^{4} - 3111793369634 T^{6} + 24669325576205464 T^{8} - 3111793369634 p^{6} T^{10} + 309658861 p^{12} T^{12} - 24014 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
23 \( ( 1 - 25876 T^{2} + 426053944 T^{4} - 8831082844 p^{2} T^{6} + 59132587818952942 T^{8} - 8831082844 p^{8} T^{10} + 426053944 p^{12} T^{12} - 25876 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
29 \( ( 1 - 2138 p T^{2} + 2900130409 T^{4} - 97031213045314 T^{6} + 2790721647797586244 T^{8} - 97031213045314 p^{6} T^{10} + 2900130409 p^{12} T^{12} - 2138 p^{19} T^{14} + p^{24} T^{16} )^{2} \)
31 \( ( 1 - 24872 T^{2} + 1564717906 T^{4} - 74785801641248 T^{6} + 1436327591399410627 T^{8} - 74785801641248 p^{6} T^{10} + 1564717906 p^{12} T^{12} - 24872 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
37 \( ( 1 + 38 T + 93295 T^{2} - 8312074 T^{3} + 3895802698 T^{4} - 8312074 p^{3} T^{5} + 93295 p^{6} T^{6} + 38 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
41 \( ( 1 + 311308 T^{2} + 49778800408 T^{4} + 5249339160024580 T^{6} + \)\(41\!\cdots\!42\)\( T^{8} + 5249339160024580 p^{6} T^{10} + 49778800408 p^{12} T^{12} + 311308 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
43 \( ( 1 - 352 T + 171577 T^{2} - 15193816 T^{3} + 8385609238 T^{4} - 15193816 p^{3} T^{5} + 171577 p^{6} T^{6} - 352 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
47 \( ( 1 + 307672 T^{2} + 59192427700 T^{4} + 8203813375455880 T^{6} + \)\(95\!\cdots\!02\)\( T^{8} + 8203813375455880 p^{6} T^{10} + 59192427700 p^{12} T^{12} + 307672 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
53 \( ( 1 - 938962 T^{2} + 414072874153 T^{4} - 111697887842318626 T^{6} + \)\(20\!\cdots\!64\)\( T^{8} - 111697887842318626 p^{6} T^{10} + 414072874153 p^{12} T^{12} - 938962 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
59 \( ( 1 + 937618 T^{2} + 446788175977 T^{4} + 144979784390319226 T^{6} + 9924470741161059460 p^{2} T^{8} + 144979784390319226 p^{6} T^{10} + 446788175977 p^{12} T^{12} + 937618 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
61 \( ( 1 - 1721996 T^{2} + 1317169449928 T^{4} - 584519836012957892 T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - 584519836012957892 p^{6} T^{10} + 1317169449928 p^{12} T^{12} - 1721996 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
67 \( ( 1 - 764 T + 857359 T^{2} - 499632752 T^{3} + 364904739412 T^{4} - 499632752 p^{3} T^{5} + 857359 p^{6} T^{6} - 764 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
71 \( ( 1 - 1856440 T^{2} + 1761671174068 T^{4} - 1073069689503907240 T^{6} + \)\(45\!\cdots\!66\)\( T^{8} - 1073069689503907240 p^{6} T^{10} + 1761671174068 p^{12} T^{12} - 1856440 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
73 \( ( 1 - 1477226 T^{2} + 830291257153 T^{4} - 215967978098523506 T^{6} + \)\(44\!\cdots\!88\)\( T^{8} - 215967978098523506 p^{6} T^{10} + 830291257153 p^{12} T^{12} - 1477226 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
79 \( ( 1 - 182 T + 1336582 T^{2} - 95923856 T^{3} + 833036352655 T^{4} - 95923856 p^{3} T^{5} + 1336582 p^{6} T^{6} - 182 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
83 \( ( 1 + 2984326 T^{2} + 4434375721357 T^{4} + 4272111895669322434 T^{6} + \)\(28\!\cdots\!68\)\( T^{8} + 4272111895669322434 p^{6} T^{10} + 4434375721357 p^{12} T^{12} + 2984326 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
89 \( ( 1 + 2563264 T^{2} + 3294847419004 T^{4} + 2897405815989956416 T^{6} + \)\(21\!\cdots\!94\)\( T^{8} + 2897405815989956416 p^{6} T^{10} + 3294847419004 p^{12} T^{12} + 2563264 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
97 \( ( 1 - 2165930 T^{2} + 2100306931633 T^{4} - 316975242262618658 T^{6} - \)\(53\!\cdots\!72\)\( T^{8} - 316975242262618658 p^{6} T^{10} + 2100306931633 p^{12} T^{12} - 2165930 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
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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

−1.91515047850102293590651896082, −1.83349270654882792014818732261, −1.82622261130347503317429335828, −1.79764882588534346391725491270, −1.76345992364009419858013169138, −1.72169526328882687614575386519, −1.61999157310109297167317342597, −1.51606060253736678874220893033, −1.39843734543465481407212898447, −1.25352471164592928184611347040, −1.11728573211078150822965351376, −0.969201039364374847722724409065, −0.879635653947831141743409782122, −0.849282511522751682816093721929, −0.847096123049882555604133647282, −0.66654686488865236829024603243, −0.64858871217544084290565973881, −0.60819333770907363965458199966, −0.53241168227028838654452383535, −0.53067611549071209338045392124, −0.47056289707358519096659551266, −0.46285627123880164551955543547, −0.34770298012381141792434185491, −0.18402669851734172610322809057, −0.13303439668573769440274856887, 0.13303439668573769440274856887, 0.18402669851734172610322809057, 0.34770298012381141792434185491, 0.46285627123880164551955543547, 0.47056289707358519096659551266, 0.53067611549071209338045392124, 0.53241168227028838654452383535, 0.60819333770907363965458199966, 0.64858871217544084290565973881, 0.66654686488865236829024603243, 0.847096123049882555604133647282, 0.849282511522751682816093721929, 0.879635653947831141743409782122, 0.969201039364374847722724409065, 1.11728573211078150822965351376, 1.25352471164592928184611347040, 1.39843734543465481407212898447, 1.51606060253736678874220893033, 1.61999157310109297167317342597, 1.72169526328882687614575386519, 1.76345992364009419858013169138, 1.79764882588534346391725491270, 1.82622261130347503317429335828, 1.83349270654882792014818732261, 1.91515047850102293590651896082

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

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