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

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Normalization:  

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

Graph of the $Z$-function along the critical line