# 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 $0$

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## 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