Properties

Label 2-85176-1.1-c1-0-44
Degree $2$
Conductor $85176$
Sign $-1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 7-s − 3·11-s − 17-s + 19-s − 7·23-s + 4·25-s + 5·29-s − 6·31-s − 3·35-s + 3·37-s − 2·41-s + 43-s + 49-s − 2·53-s − 9·55-s + 10·59-s + 11·61-s − 8·67-s + 2·71-s + 7·73-s + 3·77-s + 10·79-s + 14·83-s − 3·85-s − 2·89-s + 3·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.242·17-s + 0.229·19-s − 1.45·23-s + 4/5·25-s + 0.928·29-s − 1.07·31-s − 0.507·35-s + 0.493·37-s − 0.312·41-s + 0.152·43-s + 1/7·49-s − 0.274·53-s − 1.21·55-s + 1.30·59-s + 1.40·61-s − 0.977·67-s + 0.237·71-s + 0.819·73-s + 0.341·77-s + 1.12·79-s + 1.53·83-s − 0.325·85-s − 0.211·89-s + 0.307·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.13953896297697, −13.55757761295488, −13.26514126910409, −12.88486313269670, −12.19836265073858, −11.84646470923645, −11.05164267335936, −10.54305035009121, −10.19136751985083, −9.695889885511585, −9.313303792682402, −8.739105705596266, −7.998884620403203, −7.760426474316571, −6.792885917639906, −6.535543990070068, −5.892716323742071, −5.392438670652418, −5.070647935817638, −4.177147469215715, −3.628458228852332, −2.793728147508169, −2.303146067550825, −1.843805089578930, −0.9288437439172183, 0, 0.9288437439172183, 1.843805089578930, 2.303146067550825, 2.793728147508169, 3.628458228852332, 4.177147469215715, 5.070647935817638, 5.392438670652418, 5.892716323742071, 6.535543990070068, 6.792885917639906, 7.760426474316571, 7.998884620403203, 8.739105705596266, 9.313303792682402, 9.695889885511585, 10.19136751985083, 10.54305035009121, 11.05164267335936, 11.84646470923645, 12.19836265073858, 12.88486313269670, 13.26514126910409, 13.55757761295488, 14.13953896297697

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