Properties

Label 2-85176-1.1-c1-0-38
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 − 4·11-s + 6·17-s + 7·19-s + 23-s + 4·25-s − 29-s + 7·31-s + 3·35-s + 12·37-s − 6·41-s − 11·43-s − 3·47-s + 49-s + 3·53-s + 12·55-s + 12·59-s − 4·61-s − 6·67-s − 12·71-s + 11·73-s + 4·77-s − 15·79-s − 15·83-s − 18·85-s + 11·89-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s − 1.20·11-s + 1.45·17-s + 1.60·19-s + 0.208·23-s + 4/5·25-s − 0.185·29-s + 1.25·31-s + 0.507·35-s + 1.97·37-s − 0.937·41-s − 1.67·43-s − 0.437·47-s + 1/7·49-s + 0.412·53-s + 1.61·55-s + 1.56·59-s − 0.512·61-s − 0.733·67-s − 1.42·71-s + 1.28·73-s + 0.455·77-s − 1.68·79-s − 1.64·83-s − 1.95·85-s + 1.16·89-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: $\chi_{85176} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.20143998806136, −13.57824032057809, −13.13029739789740, −12.76431709806207, −11.95648194567500, −11.71251974853063, −11.51312569444392, −10.64358262589336, −10.07734039718953, −9.887681769912056, −9.212673481311172, −8.273853624143336, −8.192456682224115, −7.554179617282149, −7.301983067240725, −6.590054714848047, −5.848316527111885, −5.298203851402559, −4.878203992892109, −4.149948778326498, −3.549131716415440, −2.969466915616649, −2.730673266627711, −1.443471153827601, −0.7890567236838920, 0, 0.7890567236838920, 1.443471153827601, 2.730673266627711, 2.969466915616649, 3.549131716415440, 4.149948778326498, 4.878203992892109, 5.298203851402559, 5.848316527111885, 6.590054714848047, 7.301983067240725, 7.554179617282149, 8.192456682224115, 8.273853624143336, 9.212673481311172, 9.887681769912056, 10.07734039718953, 10.64358262589336, 11.51312569444392, 11.71251974853063, 11.95648194567500, 12.76431709806207, 13.13029739789740, 13.57824032057809, 14.20143998806136

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