Properties

Label 2-5265-1.1-c1-0-157
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  − 1.63·2-s + 0.670·4-s + 5-s + 2.13·7-s + 2.17·8-s − 1.63·10-s + 0.0526·11-s − 13-s − 3.48·14-s − 4.89·16-s + 2.48·17-s + 2.13·19-s + 0.670·20-s − 0.0860·22-s − 4.93·23-s + 25-s + 1.63·26-s + 1.42·28-s + 2.48·29-s − 8.16·31-s + 3.64·32-s − 4.05·34-s + 2.13·35-s − 1.11·37-s − 3.48·38-s + 2.17·40-s + 5.47·41-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.335·4-s + 0.447·5-s + 0.805·7-s + 0.768·8-s − 0.516·10-s + 0.0158·11-s − 0.277·13-s − 0.930·14-s − 1.22·16-s + 0.601·17-s + 0.488·19-s + 0.149·20-s − 0.0183·22-s − 1.02·23-s + 0.200·25-s + 0.320·26-s + 0.269·28-s + 0.461·29-s − 1.46·31-s + 0.644·32-s − 0.695·34-s + 0.360·35-s − 0.184·37-s − 0.564·38-s + 0.343·40-s + 0.854·41-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 \( 1 + 1.63T + 2T^{2} \)
7 \( 1 - 2.13T + 7T^{2} \)
11 \( 1 - 0.0526T + 11T^{2} \)
17 \( 1 - 2.48T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 + 4.93T + 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 + 8.16T + 31T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 + 9.47T + 43T^{2} \)
47 \( 1 + 9.76T + 47T^{2} \)
53 \( 1 - 3.64T + 53T^{2} \)
59 \( 1 + 7.49T + 59T^{2} \)
61 \( 1 + 5.78T + 61T^{2} \)
67 \( 1 + 6.23T + 67T^{2} \)
71 \( 1 + 2.50T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + 0.489T + 83T^{2} \)
89 \( 1 - 4.64T + 89T^{2} \)
97 \( 1 - 7.34T + 97T^{2} \)
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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

−7.907796709597891086268106723984, −7.46765708855773424068847574463, −6.61223313271570278006276058125, −5.65625136072427784225759805551, −4.97493129900159513219845231095, −4.21653868253751991273872519267, −3.10148321570925046583942350715, −1.89361027630178017955377991202, −1.37360681828738837835402271084, 0, 1.37360681828738837835402271084, 1.89361027630178017955377991202, 3.10148321570925046583942350715, 4.21653868253751991273872519267, 4.97493129900159513219845231095, 5.65625136072427784225759805551, 6.61223313271570278006276058125, 7.46765708855773424068847574463, 7.907796709597891086268106723984

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