# Properties

 Label 2-690-69.68-c1-0-7 Degree $2$ Conductor $690$ Sign $0.265 - 0.964i$ Analytic cond. $5.50967$ Root an. cond. $2.34727$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + i·2-s + (−0.798 − 1.53i)3-s − 4-s − 5-s + (1.53 − 0.798i)6-s + 0.145i·7-s − i·8-s + (−1.72 + 2.45i)9-s − i·10-s − 3.35·11-s + (0.798 + 1.53i)12-s + 2.96·13-s − 0.145·14-s + (0.798 + 1.53i)15-s + 16-s + 5.54·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + (−0.461 − 0.887i)3-s − 0.5·4-s − 0.447·5-s + (0.627 − 0.326i)6-s + 0.0548i·7-s − 0.353i·8-s + (−0.574 + 0.818i)9-s − 0.316i·10-s − 1.01·11-s + (0.230 + 0.443i)12-s + 0.821·13-s − 0.0388·14-s + (0.206 + 0.396i)15-s + 0.250·16-s + 1.34·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $0.265 - 0.964i$ Analytic conductor: $$5.50967$$ Root analytic conductor: $$2.34727$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{690} (551, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 690,\ (\ :1/2),\ 0.265 - 0.964i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.728030 + 0.554783i$$ $$L(\frac12)$$ $$\approx$$ $$0.728030 + 0.554783i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1 + (0.798 + 1.53i)T$$
5 $$1 + T$$
23 $$1 + (3.26 - 3.51i)T$$
good7 $$1 - 0.145iT - 7T^{2}$$
11 $$1 + 3.35T + 11T^{2}$$
13 $$1 - 2.96T + 13T^{2}$$
17 $$1 - 5.54T + 17T^{2}$$
19 $$1 - 6.91iT - 19T^{2}$$
29 $$1 + 3.58iT - 29T^{2}$$
31 $$1 - 10.6T + 31T^{2}$$
37 $$1 - 6.04iT - 37T^{2}$$
41 $$1 + 8.84iT - 41T^{2}$$
43 $$1 - 8.40iT - 43T^{2}$$
47 $$1 - 9.98iT - 47T^{2}$$
53 $$1 + 3.83T + 53T^{2}$$
59 $$1 - 7.91iT - 59T^{2}$$
61 $$1 - 4.14iT - 61T^{2}$$
67 $$1 - 13.0iT - 67T^{2}$$
71 $$1 - 1.26iT - 71T^{2}$$
73 $$1 + 9.24T + 73T^{2}$$
79 $$1 + 13.2iT - 79T^{2}$$
83 $$1 - 9.71T + 83T^{2}$$
89 $$1 + 15.3T + 89T^{2}$$
97 $$1 + 11.6iT - 97T^{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

−10.56063691379474939510971417280, −9.893330684521657081007205253731, −8.336343831563333330264081742766, −7.996588413131216636507069915571, −7.29086209087977691772879119635, −5.97883297454198756344911289785, −5.72832485339738902809631433952, −4.37721889341687971446122479779, −3.02178216278145939147973395249, −1.23246340678355620718874745937, 0.60375957037601511932937144204, 2.73940417646819242699731901031, 3.66860972389113541640103778288, 4.69848860803420234953292101362, 5.44442648459586088532050062961, 6.60891346140943510408506742949, 7.976582183731300741349627499867, 8.705627281897573702070614381238, 9.700294933717804513957223754890, 10.46376088096068512152253958358