Properties

Label 2-5265-1.1-c1-0-151
Degree $2$
Conductor $5265$
Sign $-1$
Analytic cond. $42.0412$
Root an. cond. $6.48392$
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  − 0.0672·2-s − 1.99·4-s + 5-s + 2.46·7-s + 0.268·8-s − 0.0672·10-s − 3.21·11-s − 13-s − 0.165·14-s + 3.97·16-s + 4.77·17-s − 3.94·19-s − 1.99·20-s + 0.215·22-s − 4.26·23-s + 25-s + 0.0672·26-s − 4.91·28-s + 2.30·29-s − 7.63·31-s − 0.803·32-s − 0.321·34-s + 2.46·35-s + 4.87·37-s + 0.265·38-s + 0.268·40-s + 2.53·41-s + ⋯
L(s)  = 1  − 0.0475·2-s − 0.997·4-s + 0.447·5-s + 0.931·7-s + 0.0949·8-s − 0.0212·10-s − 0.968·11-s − 0.277·13-s − 0.0442·14-s + 0.993·16-s + 1.15·17-s − 0.905·19-s − 0.446·20-s + 0.0460·22-s − 0.890·23-s + 0.200·25-s + 0.0131·26-s − 0.929·28-s + 0.428·29-s − 1.37·31-s − 0.142·32-s − 0.0550·34-s + 0.416·35-s + 0.801·37-s + 0.0430·38-s + 0.0424·40-s + 0.395·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5265\)    =    \(3^{4} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(42.0412\)
Root analytic conductor: \(6.48392\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5265} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5265,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + 0.0672T + 2T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
17 \( 1 - 4.77T + 17T^{2} \)
19 \( 1 + 3.94T + 19T^{2} \)
23 \( 1 + 4.26T + 23T^{2} \)
29 \( 1 - 2.30T + 29T^{2} \)
31 \( 1 + 7.63T + 31T^{2} \)
37 \( 1 - 4.87T + 37T^{2} \)
41 \( 1 - 2.53T + 41T^{2} \)
43 \( 1 + 4.56T + 43T^{2} \)
47 \( 1 - 6.26T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 2.85T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 7.79T + 71T^{2} \)
73 \( 1 - 2.26T + 73T^{2} \)
79 \( 1 - 9.13T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 0.966T + 89T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−7.88786773010059191278322039101, −7.45696464576134128430086586020, −6.16707839078175403336969484995, −5.54411183763990869734194655313, −4.92864024308028558657326721437, −4.29747006405585163582143007909, −3.34547425583030082332063418201, −2.28180131471235831962652773856, −1.33019418498566178546759213681, 0, 1.33019418498566178546759213681, 2.28180131471235831962652773856, 3.34547425583030082332063418201, 4.29747006405585163582143007909, 4.92864024308028558657326721437, 5.54411183763990869734194655313, 6.16707839078175403336969484995, 7.45696464576134128430086586020, 7.88786773010059191278322039101

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