Properties

 Degree $2$ Conductor $2015$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual yes Analytic rank $0$

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Dirichlet series

 L(s)  = 1 − 0.241·2-s + 0.709·3-s − 0.941·4-s − 5-s − 0.170·6-s − 1.77·7-s + 0.468·8-s − 0.497·9-s + 0.241·10-s + 1.13·11-s − 0.667·12-s − 13-s + 0.426·14-s − 0.709·15-s + 0.829·16-s + 1.94·17-s + 0.119·18-s + 0.941·20-s − 1.25·21-s − 0.273·22-s + 1.49·23-s + 0.332·24-s + 25-s + 0.241·26-s − 1.06·27-s + 1.66·28-s + 0.170·30-s + ⋯
 L(s)  = 1 − 0.241·2-s + 0.709·3-s − 0.941·4-s − 5-s − 0.170·6-s − 1.77·7-s + 0.468·8-s − 0.497·9-s + 0.241·10-s + 1.13·11-s − 0.667·12-s − 13-s + 0.426·14-s − 0.709·15-s + 0.829·16-s + 1.94·17-s + 0.119·18-s + 0.941·20-s − 1.25·21-s − 0.273·22-s + 1.49·23-s + 0.332·24-s + 25-s + 0.241·26-s − 1.06·27-s + 1.66·28-s + 0.170·30-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$2015$$    =    $$5 \cdot 13 \cdot 31$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{2015} (2014, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2015,\ (\ :0),\ 1)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6754621010$$ $$L(\frac12)$$ $$\approx$$ $$0.6754621010$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + T$$
13 $$1 + T$$
31 $$1 - T$$
good2 $$1 + 0.241T + T^{2}$$
3 $$1 - 0.709T + T^{2}$$
7 $$1 + 1.77T + T^{2}$$
11 $$1 - 1.13T + T^{2}$$
17 $$1 - 1.94T + T^{2}$$
19 $$1 - T^{2}$$
23 $$1 - 1.49T + T^{2}$$
29 $$1 - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 0.241T + T^{2}$$
47 $$1 - 1.49T + T^{2}$$
53 $$1 + 1.13T + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + 1.13T + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - 1.77T + T^{2}$$
97 $$1 - 0.709T + 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

−9.231128206993928047183578181308, −8.787559286829685999830269067092, −7.81347785847448157897884612512, −7.26801640739764489979896167791, −6.28331138789285457717973570435, −5.25504363970816139280431196278, −4.19273826275226385932867657323, −3.32605393727601151168199851474, −3.02315865078478646114561752633, −0.812816826684260909474368882356, 0.812816826684260909474368882356, 3.02315865078478646114561752633, 3.32605393727601151168199851474, 4.19273826275226385932867657323, 5.25504363970816139280431196278, 6.28331138789285457717973570435, 7.26801640739764489979896167791, 7.81347785847448157897884612512, 8.787559286829685999830269067092, 9.231128206993928047183578181308