Properties

Label 2-5290-1.1-c1-0-76
Degree $2$
Conductor $5290$
Sign $-1$
Analytic cond. $42.2408$
Root an. cond. $6.49929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.92·3-s + 4-s − 5-s − 2.92·6-s − 4.55·7-s + 8-s + 5.55·9-s − 10-s + 0.925·11-s − 2.92·12-s + 2.55·13-s − 4.55·14-s + 2.92·15-s + 16-s − 6.92·17-s + 5.55·18-s + 3.29·19-s − 20-s + 13.3·21-s + 0.925·22-s − 2.92·24-s + 25-s + 2.55·26-s − 7.48·27-s − 4.55·28-s − 1.26·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.68·3-s + 0.5·4-s − 0.447·5-s − 1.19·6-s − 1.72·7-s + 0.353·8-s + 1.85·9-s − 0.316·10-s + 0.279·11-s − 0.844·12-s + 0.709·13-s − 1.21·14-s + 0.755·15-s + 0.250·16-s − 1.67·17-s + 1.31·18-s + 0.755·19-s − 0.223·20-s + 2.90·21-s + 0.197·22-s − 0.597·24-s + 0.200·25-s + 0.501·26-s − 1.44·27-s − 0.861·28-s − 0.234·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5290\)    =    \(2 \cdot 5 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(42.2408\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5290} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5290,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 \)
good3 \( 1 + 2.92T + 3T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
11 \( 1 - 0.925T + 11T^{2} \)
13 \( 1 - 2.55T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 3.29T + 19T^{2} \)
29 \( 1 + 1.26T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 9.70T + 37T^{2} \)
41 \( 1 + 0.925T + 41T^{2} \)
43 \( 1 - 1.26T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 3.11T + 53T^{2} \)
59 \( 1 + 7.26T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 7.85T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 9.26T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 0.585T + 83T^{2} \)
89 \( 1 + 9.11T + 89T^{2} \)
97 \( 1 - 3.22T + 97T^{2} \)
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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

−7.31187067804661166158739015584, −6.73727297102760109122138180987, −6.38480758613488357047713353199, −5.77899586591440957168400356729, −4.98185948304660248890110788684, −4.17107172911167583839091045844, −3.59247245033896319884606821920, −2.54979589165270434288103267782, −1.03102244215558555122025674850, 0, 1.03102244215558555122025674850, 2.54979589165270434288103267782, 3.59247245033896319884606821920, 4.17107172911167583839091045844, 4.98185948304660248890110788684, 5.77899586591440957168400356729, 6.38480758613488357047713353199, 6.73727297102760109122138180987, 7.31187067804661166158739015584

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