Properties

 Label 2-1216-1.1-c3-0-53 Degree $2$ Conductor $1216$ Sign $-1$ Analytic cond. $71.7463$ Root an. cond. $8.47032$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

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Dirichlet series

 L(s)  = 1 − 3-s − 17.2·5-s − 12.3·7-s − 26·9-s + 39.2·11-s − 18.9·13-s + 17.2·15-s + 100.·17-s + 19·19-s + 12.3·21-s + 158.·23-s + 173.·25-s + 53·27-s − 141.·29-s + 215.·31-s − 39.2·33-s + 213.·35-s − 209.·37-s + 18.9·39-s + 276.·41-s − 166.·43-s + 449.·45-s − 60.5·47-s − 190.·49-s − 100.·51-s + 76.2·53-s − 679.·55-s + ⋯
 L(s)  = 1 − 0.192·3-s − 1.54·5-s − 0.667·7-s − 0.962·9-s + 1.07·11-s − 0.403·13-s + 0.297·15-s + 1.42·17-s + 0.229·19-s + 0.128·21-s + 1.43·23-s + 1.39·25-s + 0.377·27-s − 0.904·29-s + 1.24·31-s − 0.207·33-s + 1.03·35-s − 0.932·37-s + 0.0777·39-s + 1.05·41-s − 0.591·43-s + 1.48·45-s − 0.187·47-s − 0.554·49-s − 0.274·51-s + 0.197·53-s − 1.66·55-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $-1$ Analytic conductor: $$71.7463$$ Root analytic conductor: $$8.47032$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1216,\ (\ :3/2),\ -1)$$

Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 - 19T$$
good3 $$1 + T + 27T^{2}$$
5 $$1 + 17.2T + 125T^{2}$$
7 $$1 + 12.3T + 343T^{2}$$
11 $$1 - 39.2T + 1.33e3T^{2}$$
13 $$1 + 18.9T + 2.19e3T^{2}$$
17 $$1 - 100.T + 4.91e3T^{2}$$
23 $$1 - 158.T + 1.21e4T^{2}$$
29 $$1 + 141.T + 2.43e4T^{2}$$
31 $$1 - 215.T + 2.97e4T^{2}$$
37 $$1 + 209.T + 5.06e4T^{2}$$
41 $$1 - 276.T + 6.89e4T^{2}$$
43 $$1 + 166.T + 7.95e4T^{2}$$
47 $$1 + 60.5T + 1.03e5T^{2}$$
53 $$1 - 76.2T + 1.48e5T^{2}$$
59 $$1 + 575.T + 2.05e5T^{2}$$
61 $$1 - 91.0T + 2.26e5T^{2}$$
67 $$1 - 444.T + 3.00e5T^{2}$$
71 $$1 + 943.T + 3.57e5T^{2}$$
73 $$1 + 569.T + 3.89e5T^{2}$$
79 $$1 - 336.T + 4.93e5T^{2}$$
83 $$1 + 603.T + 5.71e5T^{2}$$
89 $$1 - 1.63e3T + 7.04e5T^{2}$$
97 $$1 - 319.T + 9.12e5T^{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

−8.917893205469915773158969755520, −8.083738722835255727420789698462, −7.33711514859523866606934098749, −6.56410156099193031788154077670, −5.55413727175344320027893379523, −4.55136195158787050176654012662, −3.49115604823327394888079237802, −3.02263423017133149104774076181, −1.04993588526087211607630818539, 0, 1.04993588526087211607630818539, 3.02263423017133149104774076181, 3.49115604823327394888079237802, 4.55136195158787050176654012662, 5.55413727175344320027893379523, 6.56410156099193031788154077670, 7.33711514859523866606934098749, 8.083738722835255727420789698462, 8.917893205469915773158969755520