Properties

Label 32-42e32-1.1-c3e16-0-0
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  + 520·25-s − 512·37-s + 64·43-s + 2.33e3·67-s − 1.79e3·79-s + 7.74e3·109-s − 8.99e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4.15·25-s − 2.27·37-s + 0.226·43-s + 4.25·67-s − 2.55·79-s + 6.80·109-s − 6.75·121-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 + 6.73·169-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

\(L(2)\) \(\approx\) \(0.02797253389\)
\(L(\frac12)\) \(\approx\) \(0.02797253389\)
\(L(\frac{5}{2})\) 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 \( ( 1 - 52 p T^{2} + 32698 T^{4} - 189904 p T^{6} - 82241021 T^{8} - 189904 p^{7} T^{10} + 32698 p^{12} T^{12} - 52 p^{19} T^{14} + p^{24} T^{16} )^{2} \)
11 \( ( 1 + 2248 T^{2} + 3281943 T^{4} + 2248 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
13 \( ( 1 - 3700 T^{2} + 10479510 T^{4} - 3700 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
17 \( ( 1 - 3284 T^{2} + 26596522 T^{4} + 210461721136 T^{6} - 592922509169069 T^{8} + 210461721136 p^{6} T^{10} + 26596522 p^{12} T^{12} - 3284 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
19 \( ( 1 + 5644 T^{2} + 46858618 T^{4} - 1705639376 p^{2} T^{6} - 26755722893 p^{4} T^{8} - 1705639376 p^{8} T^{10} + 46858618 p^{12} T^{12} + 5644 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
23 \( ( 1 + 26672 T^{2} + 264302110 T^{4} + 4028050675712 T^{6} + 72665872353254179 T^{8} + 4028050675712 p^{6} T^{10} + 264302110 p^{12} T^{12} + 26672 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
29 \( ( 1 - 14432 T^{2} + 1444578 p^{2} T^{4} - 14432 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
31 \( ( 1 + 81148 T^{2} + 3232418698 T^{4} + 128016799996912 T^{6} + 4607914669537756051 T^{8} + 128016799996912 p^{6} T^{10} + 3232418698 p^{12} T^{12} + 81148 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
37 \( ( 1 + 128 T - 59210 T^{2} - 3291136 T^{3} + 2318827675 T^{4} - 3291136 p^{3} T^{5} - 59210 p^{6} T^{6} + 128 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
41 \( ( 1 - 52876 T^{2} + 5788532934 T^{4} - 52876 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
43 \( ( 1 - 8 T + 138330 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{8} \)
47 \( ( 1 - 7676 T^{2} - 21503853494 T^{4} - 33343100912 T^{6} + \)\(34\!\cdots\!55\)\( T^{8} - 33343100912 p^{6} T^{10} - 21503853494 p^{12} T^{12} - 7676 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
53 \( ( 1 + 272480 T^{2} + 26044792174 T^{4} + 1054997864560640 T^{6} + \)\(18\!\cdots\!35\)\( T^{8} + 1054997864560640 p^{6} T^{10} + 26044792174 p^{12} T^{12} + 272480 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
59 \( ( 1 - 718604 T^{2} + 303597445978 T^{4} - 92292608059323824 T^{6} + \)\(21\!\cdots\!07\)\( T^{8} - 92292608059323824 p^{6} T^{10} + 303597445978 p^{12} T^{12} - 718604 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
61 \( ( 1 + 197524 T^{2} + 30085369210 T^{4} - 18589060152106544 T^{6} - \)\(45\!\cdots\!01\)\( T^{8} - 18589060152106544 p^{6} T^{10} + 30085369210 p^{12} T^{12} + 197524 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
67 \( ( 1 - 584 T - 332486 T^{2} - 42057344 T^{3} + 260623417867 T^{4} - 42057344 p^{3} T^{5} - 332486 p^{6} T^{6} - 584 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
71 \( ( 1 - 1100336 T^{2} + 549200776866 T^{4} - 1100336 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
73 \( ( 1 + 1092484 T^{2} + 592523440426 T^{4} + 325920093227453968 T^{6} + \)\(15\!\cdots\!95\)\( T^{8} + 325920093227453968 p^{6} T^{10} + 592523440426 p^{12} T^{12} + 1092484 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
79 \( ( 1 + 448 T - 716318 T^{2} - 30937088 T^{3} + 498886993507 T^{4} - 30937088 p^{3} T^{5} - 716318 p^{6} T^{6} + 448 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
83 \( ( 1 + 856556 T^{2} + 654365717334 T^{4} + 856556 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
89 \( ( 1 - 2316596 T^{2} + 3070147511818 T^{4} - 3017382352062767696 T^{6} + \)\(23\!\cdots\!07\)\( T^{8} - 3017382352062767696 p^{6} T^{10} + 3070147511818 p^{12} T^{12} - 2316596 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
97 \( ( 1 - 2834596 T^{2} + 3657520887750 T^{4} - 2834596 p^{6} T^{6} + p^{12} T^{8} )^{4} \)
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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

−1.91164279962789596997821125104, −1.79779270357378446965413261690, −1.76674710634667648707852151611, −1.75868062492729732861754804425, −1.74241760064494907574554902060, −1.59218734331027296567643208873, −1.53723352731498877554564042926, −1.29972413216267098286677776186, −1.20956429166522217343968267048, −1.18991839842744878499863440885, −1.17907958428480659232976867141, −1.13592653712833560041401159750, −1.07769571880923767515591801550, −1.04258562139298334236928713965, −0.846670907497015282117748385571, −0.836067506688269267161734674127, −0.57832588315220470545116783569, −0.55799323611573059238147257374, −0.52773734315153489706875140845, −0.52154204556430233327137362048, −0.49788572916356760919049940646, −0.36870551934504908965761404342, −0.28821513830757652770264735097, −0.10762165277977245011961541333, −0.00317186263283838484808032871, 0.00317186263283838484808032871, 0.10762165277977245011961541333, 0.28821513830757652770264735097, 0.36870551934504908965761404342, 0.49788572916356760919049940646, 0.52154204556430233327137362048, 0.52773734315153489706875140845, 0.55799323611573059238147257374, 0.57832588315220470545116783569, 0.836067506688269267161734674127, 0.846670907497015282117748385571, 1.04258562139298334236928713965, 1.07769571880923767515591801550, 1.13592653712833560041401159750, 1.17907958428480659232976867141, 1.18991839842744878499863440885, 1.20956429166522217343968267048, 1.29972413216267098286677776186, 1.53723352731498877554564042926, 1.59218734331027296567643208873, 1.74241760064494907574554902060, 1.75868062492729732861754804425, 1.76674710634667648707852151611, 1.79779270357378446965413261690, 1.91164279962789596997821125104

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

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