# Properties

 Label 2-507-39.32-c1-0-19 Degree $2$ Conductor $507$ Sign $0.281 - 0.959i$ Analytic cond. $4.04841$ Root an. cond. $2.01206$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−2.31 + 0.619i)2-s + (1.64 + 0.529i)3-s + (3.23 − 1.86i)4-s + (1.69 + 1.69i)5-s + (−4.14 − 0.202i)6-s + (0.366 − 1.36i)7-s + (−2.93 + 2.93i)8-s + (2.43 + 1.74i)9-s + (−4.96 − 2.86i)10-s + (0.453 + 1.69i)11-s + (6.31 − 1.36i)12-s + 3.38i·14-s + (1.89 + 3.68i)15-s + (1.23 − 2.13i)16-s + (1.07 + 1.85i)17-s + (−6.72 − 2.52i)18-s + ⋯
 L(s)  = 1 + (−1.63 + 0.438i)2-s + (0.952 + 0.305i)3-s + (1.61 − 0.933i)4-s + (0.757 + 0.757i)5-s + (−1.69 − 0.0826i)6-s + (0.138 − 0.516i)7-s + (−1.03 + 1.03i)8-s + (0.813 + 0.582i)9-s + (−1.56 − 0.906i)10-s + (0.136 + 0.510i)11-s + (1.82 − 0.394i)12-s + 0.904i·14-s + (0.489 + 0.952i)15-s + (0.308 − 0.533i)16-s + (0.260 + 0.450i)17-s + (−1.58 − 0.595i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $0.281 - 0.959i$ Analytic conductor: $$4.04841$$ Root analytic conductor: $$2.01206$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{507} (188, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 507,\ (\ :1/2),\ 0.281 - 0.959i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.875005 + 0.655513i$$ $$L(\frac12)$$ $$\approx$$ $$0.875005 + 0.655513i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.64 - 0.529i)T$$
13 $$1$$
good2 $$1 + (2.31 - 0.619i)T + (1.73 - i)T^{2}$$
5 $$1 + (-1.69 - 1.69i)T + 5iT^{2}$$
7 $$1 + (-0.366 + 1.36i)T + (-6.06 - 3.5i)T^{2}$$
11 $$1 + (-0.453 - 1.69i)T + (-9.52 + 5.5i)T^{2}$$
17 $$1 + (-1.07 - 1.85i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-1 - 0.267i)T + (16.4 + 9.5i)T^{2}$$
23 $$1 + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (4.79 + 2.76i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (-4.46 + 4.46i)T - 31iT^{2}$$
37 $$1 + (-6.59 + 1.76i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 + (-0.619 + 0.166i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (7.09 - 4.09i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (6.77 - 6.77i)T - 47iT^{2}$$
53 $$1 + 4.62iT - 53T^{2}$$
59 $$1 + (4.62 + 1.23i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-2.26 - 8.46i)T + (-58.0 + 33.5i)T^{2}$$
71 $$1 + (1.23 - 4.62i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + (6.09 + 6.09i)T + 73iT^{2}$$
79 $$1 - 2T + 79T^{2}$$
83 $$1 + (1.23 + 1.23i)T + 83iT^{2}$$
89 $$1 + (-2.60 - 9.70i)T + (-77.0 + 44.5i)T^{2}$$
97 $$1 + (12.5 + 3.36i)T + (84.0 + 48.5i)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

−10.56191761866180207428615104429, −9.799148065024493174149142799066, −9.583087327149695266304669654871, −8.381744172806616499897745075253, −7.70139041522317021765024169660, −6.94290116452962223740342052681, −5.99635707702748889758638225144, −4.23802177462681480915676742027, −2.67892749542393242998478622004, −1.58295154897055578782395973987, 1.14112696066653725154897812640, 2.12763078534259849622766685025, 3.25818926469442174774571190645, 5.12590690739692516227907041671, 6.51227470848891205098105328816, 7.57510087254043119625147833433, 8.418236015789392275949493728634, 8.987903023908555265873712004541, 9.551064831666355912790982314254, 10.30632358353984924217587349516