Properties

Label 2-68-68.67-c8-0-35
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

\(L(\frac{9}{2})\) \(\approx\) \(2.717948817\)
\(L(\frac12)\) \(\approx\) \(2.717948817\)
\(L(5)\) 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

−12.95079403993241973743604526274, −12.01952268736460787012497370185, −11.09832688041167133289232079705, −10.30012127154877689087320146612, −8.101553804892734313090490204813, −6.67292337689288317151431874389, −5.58814005476101746257389320942, −4.73352465627200579437177853174, −2.96030185778791903739411260963, −1.01528018976289287587303516040, 1.01528018976289287587303516040, 2.96030185778791903739411260963, 4.73352465627200579437177853174, 5.58814005476101746257389320942, 6.67292337689288317151431874389, 8.101553804892734313090490204813, 10.30012127154877689087320146612, 11.09832688041167133289232079705, 12.01952268736460787012497370185, 12.95079403993241973743604526274

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