Properties

Label 32-42e32-1.1-c1e16-0-10
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $2.40110\times 10^{18}$
Root an. cond. $3.75308$
Motivic weight $1$
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  + 32·25-s − 32·37-s + 64·43-s − 32·67-s + 32·109-s − 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 32/5·25-s − 5.26·37-s + 9.75·43-s − 3.90·67-s + 3.06·109-s − 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(2.40110\times 10^{18}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(103.8251306\)
\(L(\frac12)\) \(\approx\) \(103.8251306\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 16 T^{2} + 144 T^{4} - 992 T^{6} + 5519 T^{8} - 992 p^{2} T^{10} + 144 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
13 \( ( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 32 T^{2} + 352 T^{4} - 3008 T^{6} + 72127 T^{8} - 3008 p^{2} T^{10} + 352 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 + 60 T^{2} + 2010 T^{4} + 52080 T^{6} + 1098179 T^{8} + 52080 p^{2} T^{10} + 2010 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 4 T^{2} + 106 T^{4} - 4592 T^{6} - 280205 T^{8} - 4592 p^{2} T^{10} + 106 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 68 T^{2} + 2326 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 44 T^{2} - 438 T^{4} + 19888 T^{6} + 2851859 T^{8} + 19888 p^{2} T^{10} - 438 p^{4} T^{12} + 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 16 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 + 16 T^{2} - 992 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
43 \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{8} \)
47 \( ( 1 + 20 T^{2} - 2550 T^{4} - 29360 T^{6} + 2939219 T^{8} - 29360 p^{2} T^{10} - 2550 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 + 48 T^{2} + 718 T^{4} - 193536 T^{6} - 11847021 T^{8} - 193536 p^{2} T^{10} + 718 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 36 T^{2} - 4422 T^{4} - 44784 T^{6} + 16066787 T^{8} - 44784 p^{2} T^{10} - 4422 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 80 T^{2} + 2160 T^{4} - 256160 T^{6} - 19682641 T^{8} - 256160 p^{2} T^{10} + 2160 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 8 T - 14 T^{2} - 448 T^{3} - 2693 T^{4} - 448 p T^{5} - 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 20 T^{2} + 8134 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
73 \( ( 1 + 144 T^{2} + 9312 T^{4} + 110304 T^{6} - 8392609 T^{8} + 110304 p^{2} T^{10} + 9312 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 30 T^{2} - 5341 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
83 \( ( 1 + 204 T^{2} + 22134 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
89 \( ( 1 - 144 T^{2} + 10368 T^{4} + 788256 T^{6} - 119496673 T^{8} + 788256 p^{2} T^{10} + 10368 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 240 T^{2} + 28800 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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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

−2.35714739432557598647573127280, −2.35340891286035859868470502450, −2.20935580831365729700438002185, −2.16484007135508798724745149521, −2.14739706257419963313101573116, −1.93006058262466387980348175304, −1.82546186454087569203479751622, −1.75657899273997806460310302192, −1.73590163173177323597998725148, −1.66832355030722844252371148332, −1.50373719802504994164737598278, −1.49092794506067509336677546823, −1.40209614424749763512414852280, −1.30455009581083657124583315298, −1.13216684142370967470789543172, −1.10145819441073998466404539947, −1.02293797138251378167843024267, −0.957363309942025848511252379754, −0.838964674474477888086763478838, −0.73287434612554404919906838833, −0.51694433911682296203608214111, −0.51578204991273470779417867392, −0.42135657984947200178975323908, −0.39638261349962506714130076292, −0.30584576872047592645761457221, 0.30584576872047592645761457221, 0.39638261349962506714130076292, 0.42135657984947200178975323908, 0.51578204991273470779417867392, 0.51694433911682296203608214111, 0.73287434612554404919906838833, 0.838964674474477888086763478838, 0.957363309942025848511252379754, 1.02293797138251378167843024267, 1.10145819441073998466404539947, 1.13216684142370967470789543172, 1.30455009581083657124583315298, 1.40209614424749763512414852280, 1.49092794506067509336677546823, 1.50373719802504994164737598278, 1.66832355030722844252371148332, 1.73590163173177323597998725148, 1.75657899273997806460310302192, 1.82546186454087569203479751622, 1.93006058262466387980348175304, 2.14739706257419963313101573116, 2.16484007135508798724745149521, 2.20935580831365729700438002185, 2.35340891286035859868470502450, 2.35714739432557598647573127280

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

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