Properties

Label 2-85176-1.1-c1-0-19
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 7-s + 6·11-s + 8·17-s + 19-s − 23-s + 4·25-s + 5·29-s − 3·31-s + 3·35-s − 12·37-s + 10·41-s − 11·43-s − 3·47-s + 49-s + 53-s − 18·55-s − 8·59-s + 2·61-s − 2·67-s + 8·71-s − 11·73-s − 6·77-s + 13·79-s − 7·83-s − 24·85-s + 7·89-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 1.80·11-s + 1.94·17-s + 0.229·19-s − 0.208·23-s + 4/5·25-s + 0.928·29-s − 0.538·31-s + 0.507·35-s − 1.97·37-s + 1.56·41-s − 1.67·43-s − 0.437·47-s + 1/7·49-s + 0.137·53-s − 2.42·55-s − 1.04·59-s + 0.256·61-s − 0.244·67-s + 0.949·71-s − 1.28·73-s − 0.683·77-s + 1.46·79-s − 0.768·83-s − 2.60·85-s + 0.741·89-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: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.907189293\)
\(L(\frac12)\) \(\approx\) \(1.907189293\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.99066367811177, −13.60358915775903, −12.63382511494794, −12.27479751192237, −12.04243979457844, −11.60127053226991, −11.10025072958284, −10.41397012563409, −9.922337468264269, −9.414539643101321, −8.866678896599490, −8.337200105826276, −7.859288922264846, −7.286173369762304, −6.883806829811790, −6.295997845065853, −5.715238975236019, −5.013362143897317, −4.401400623954933, −3.737495935970008, −3.467694942949532, −3.017444733632863, −1.811463990429315, −1.195897375335624, −0.5014688655162279, 0.5014688655162279, 1.195897375335624, 1.811463990429315, 3.017444733632863, 3.467694942949532, 3.737495935970008, 4.401400623954933, 5.013362143897317, 5.715238975236019, 6.295997845065853, 6.883806829811790, 7.286173369762304, 7.859288922264846, 8.337200105826276, 8.866678896599490, 9.414539643101321, 9.922337468264269, 10.41397012563409, 11.10025072958284, 11.60127053226991, 12.04243979457844, 12.27479751192237, 12.63382511494794, 13.60358915775903, 13.99066367811177

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