Properties

Label 2-5265-1.1-c1-0-115
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  + 0.655·2-s − 1.57·4-s − 5-s − 0.777·7-s − 2.33·8-s − 0.655·10-s − 4.06·11-s − 13-s − 0.509·14-s + 1.60·16-s + 5.29·17-s + 6.65·19-s + 1.57·20-s − 2.66·22-s + 2.30·23-s + 25-s − 0.655·26-s + 1.22·28-s + 6.38·29-s − 8.94·31-s + 5.73·32-s + 3.46·34-s + 0.777·35-s − 2.80·37-s + 4.35·38-s + 2.33·40-s + 7.60·41-s + ⋯
L(s)  = 1  + 0.463·2-s − 0.785·4-s − 0.447·5-s − 0.293·7-s − 0.827·8-s − 0.207·10-s − 1.22·11-s − 0.277·13-s − 0.136·14-s + 0.402·16-s + 1.28·17-s + 1.52·19-s + 0.351·20-s − 0.568·22-s + 0.480·23-s + 0.200·25-s − 0.128·26-s + 0.230·28-s + 1.18·29-s − 1.60·31-s + 1.01·32-s + 0.594·34-s + 0.131·35-s − 0.461·37-s + 0.706·38-s + 0.369·40-s + 1.18·41-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: Trivial
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 \( 1 - 0.655T + 2T^{2} \)
7 \( 1 + 0.777T + 7T^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 6.65T + 19T^{2} \)
23 \( 1 - 2.30T + 23T^{2} \)
29 \( 1 - 6.38T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 + 2.80T + 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 - 8.84T + 43T^{2} \)
47 \( 1 + 4.08T + 47T^{2} \)
53 \( 1 + 4.78T + 53T^{2} \)
59 \( 1 + 4.74T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 0.307T + 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 - 7.61T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 6.08T + 97T^{2} \)
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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

−7.69525015427714863500188359113, −7.40593659118047015113484789244, −6.19206196991138590711723231538, −5.37880783636124300996622726460, −5.07216932052585768750210591241, −4.14096172707395788641469974978, −3.21205417076425851155362046226, −2.86225715158227223845418742106, −1.15735604336170687004414838404, 0, 1.15735604336170687004414838404, 2.86225715158227223845418742106, 3.21205417076425851155362046226, 4.14096172707395788641469974978, 5.07216932052585768750210591241, 5.37880783636124300996622726460, 6.19206196991138590711723231538, 7.40593659118047015113484789244, 7.69525015427714863500188359113

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