Properties

Label 2-5780-1.1-c1-0-87
Degree $2$
Conductor $5780$
Sign $-1$
Analytic cond. $46.1535$
Root an. cond. $6.79363$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s − 2·7-s + 9-s + 2·13-s + 2·15-s − 4·19-s − 4·21-s − 6·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s + 4·39-s − 6·41-s − 10·43-s + 45-s − 6·47-s − 3·49-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s − 2·63-s + 2·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5780\)    =    \(2^{2} \cdot 5 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(46.1535\)
Root analytic conductor: \(6.79363\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5780} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5780,\ (\ :1/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
17 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.082512830031549440182862089864, −7.01685578318437876150059475429, −6.38203509181244693794314477824, −5.76507277600742851293762909693, −4.75921771941889428667826381084, −3.70547400509891418049828196703, −3.34451601116632628272488100401, −2.32693600233941410331345912048, −1.69870420309408692446890199866, 0, 1.69870420309408692446890199866, 2.32693600233941410331345912048, 3.34451601116632628272488100401, 3.70547400509891418049828196703, 4.75921771941889428667826381084, 5.76507277600742851293762909693, 6.38203509181244693794314477824, 7.01685578318437876150059475429, 8.082512830031549440182862089864

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