Properties

 Label 2-28e2-1.1-c5-0-73 Degree $2$ Conductor $784$ Sign $-1$ Analytic cond. $125.740$ Root an. cond. $11.2134$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

Origins

Dirichlet series

 L(s)  = 1 − 243·9-s + 76·11-s + 4.95e3·23-s − 3.12e3·25-s + 7.28e3·29-s − 8.88e3·37-s − 1.17e4·43-s + 2.45e4·53-s − 6.93e4·67-s + 2.22e3·71-s − 8.01e4·79-s + 5.90e4·81-s − 1.84e4·99-s + 6.49e4·107-s − 2.19e5·109-s + 1.23e5·113-s + ⋯
 L(s)  = 1 − 9-s + 0.189·11-s + 1.95·23-s − 25-s + 1.60·29-s − 1.06·37-s − 0.968·43-s + 1.20·53-s − 1.88·67-s + 0.0523·71-s − 1.44·79-s + 81-s − 0.189·99-s + 0.548·107-s − 1.77·109-s + 0.907·113-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$784$$    =    $$2^{4} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$125.740$$ Root analytic conductor: $$11.2134$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 784,\ (\ :5/2),\ -1)$$

Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 + p^{5} T^{2}$$
5 $$1 + p^{5} T^{2}$$
11 $$1 - 76 T + p^{5} T^{2}$$
13 $$1 + p^{5} T^{2}$$
17 $$1 + p^{5} T^{2}$$
19 $$1 + p^{5} T^{2}$$
23 $$1 - 4952 T + p^{5} T^{2}$$
29 $$1 - 7282 T + p^{5} T^{2}$$
31 $$1 + p^{5} T^{2}$$
37 $$1 + 8886 T + p^{5} T^{2}$$
41 $$1 + p^{5} T^{2}$$
43 $$1 + 11748 T + p^{5} T^{2}$$
47 $$1 + p^{5} T^{2}$$
53 $$1 - 24550 T + p^{5} T^{2}$$
59 $$1 + p^{5} T^{2}$$
61 $$1 + p^{5} T^{2}$$
67 $$1 + 69364 T + p^{5} T^{2}$$
71 $$1 - 2224 T + p^{5} T^{2}$$
73 $$1 + p^{5} T^{2}$$
79 $$1 + 80168 T + p^{5} T^{2}$$
83 $$1 + p^{5} T^{2}$$
89 $$1 + p^{5} T^{2}$$
97 $$1 + p^{5} 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

−8.927239268082452478690194704291, −8.468054977008571319468593971231, −7.34847590775182169497905549943, −6.49274819975240320581294009898, −5.54186473877240889952814440263, −4.69389896881028523496819430429, −3.42805135504980879472105824248, −2.58970239925352720994059983834, −1.22480681209890432698537454433, 0, 1.22480681209890432698537454433, 2.58970239925352720994059983834, 3.42805135504980879472105824248, 4.69389896881028523496819430429, 5.54186473877240889952814440263, 6.49274819975240320581294009898, 7.34847590775182169497905549943, 8.468054977008571319468593971231, 8.927239268082452478690194704291