Properties

Label 2-68-68.67-c8-0-56
Degree $2$
Conductor $68$
Sign $1$
Analytic cond. $27.7017$
Root an. cond. $5.26324$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 94·3-s + 256·4-s + 1.50e3·6-s − 706·7-s + 4.09e3·8-s + 2.27e3·9-s + 1.39e4·11-s + 2.40e4·12-s + 1.79e4·13-s − 1.12e4·14-s + 6.55e4·16-s + 8.35e4·17-s + 3.64e4·18-s − 6.63e4·21-s + 2.23e5·22-s − 1.90e5·23-s + 3.85e5·24-s + 3.90e5·25-s + 2.87e5·26-s − 4.02e5·27-s − 1.80e5·28-s − 1.78e6·31-s + 1.04e6·32-s + 1.31e6·33-s + 1.33e6·34-s + 5.82e5·36-s + ⋯
L(s)  = 1  + 2-s + 1.16·3-s + 4-s + 1.16·6-s − 0.294·7-s + 8-s + 0.346·9-s + 0.954·11-s + 1.16·12-s + 0.628·13-s − 0.294·14-s + 16-s + 17-s + 0.346·18-s − 0.341·21-s + 0.954·22-s − 0.679·23-s + 1.16·24-s + 25-s + 0.628·26-s − 0.758·27-s − 0.294·28-s − 1.92·31-s + 32-s + 1.10·33-s + 34-s + 0.346·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(27.7017\)
Root analytic conductor: \(5.26324\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: $\chi_{68} (67, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(5.820140426\)
\(L(\frac12)\) \(\approx\) \(5.820140426\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{4} T \)
17 \( 1 - p^{4} T \)
good3 \( 1 - 94 T + p^{8} T^{2} \)
5 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
7 \( 1 + 706 T + p^{8} T^{2} \)
11 \( 1 - 13982 T + p^{8} T^{2} \)
13 \( 1 - 17954 T + p^{8} T^{2} \)
19 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
23 \( 1 + 190018 T + p^{8} T^{2} \)
29 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
31 \( 1 + 1781506 T + p^{8} T^{2} \)
37 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
41 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
43 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
47 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
53 \( 1 - 1641314 T + p^{8} T^{2} \)
59 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
61 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
67 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
71 \( 1 - 11649854 T + p^{8} T^{2} \)
73 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
79 \( 1 + 70845826 T + p^{8} T^{2} \)
83 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
89 \( 1 + 125190718 T + p^{8} T^{2} \)
97 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
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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

−13.29911251098669914119767197868, −12.28920314217214881647056620270, −11.06047012736672639209272026050, −9.597092136559466418761650967913, −8.348894100102730294291421914675, −7.07465121020981460062024861911, −5.72853467425643044090180088388, −3.94819103849281876752317692854, −3.08497425233300462836458805228, −1.59303606450557875227022473815, 1.59303606450557875227022473815, 3.08497425233300462836458805228, 3.94819103849281876752317692854, 5.72853467425643044090180088388, 7.07465121020981460062024861911, 8.348894100102730294291421914675, 9.597092136559466418761650967913, 11.06047012736672639209272026050, 12.28920314217214881647056620270, 13.29911251098669914119767197868

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