Properties

Label 2-5265-1.1-c1-0-140
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  − 0.475·2-s − 1.77·4-s − 5-s + 2.49·7-s + 1.79·8-s + 0.475·10-s + 3.00·11-s + 13-s − 1.18·14-s + 2.69·16-s + 1.86·17-s − 6.61·19-s + 1.77·20-s − 1.42·22-s − 4.00·23-s + 25-s − 0.475·26-s − 4.42·28-s − 7.54·29-s − 1.10·31-s − 4.86·32-s − 0.885·34-s − 2.49·35-s − 10.2·37-s + 3.14·38-s − 1.79·40-s + 3.93·41-s + ⋯
L(s)  = 1  − 0.336·2-s − 0.887·4-s − 0.447·5-s + 0.943·7-s + 0.634·8-s + 0.150·10-s + 0.906·11-s + 0.277·13-s − 0.316·14-s + 0.673·16-s + 0.451·17-s − 1.51·19-s + 0.396·20-s − 0.304·22-s − 0.834·23-s + 0.200·25-s − 0.0932·26-s − 0.836·28-s − 1.40·29-s − 0.198·31-s − 0.860·32-s − 0.151·34-s − 0.421·35-s − 1.67·37-s + 0.510·38-s − 0.283·40-s + 0.614·41-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 \( 1 + 0.475T + 2T^{2} \)
7 \( 1 - 2.49T + 7T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
17 \( 1 - 1.86T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 + 4.00T + 23T^{2} \)
29 \( 1 + 7.54T + 29T^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 3.93T + 41T^{2} \)
43 \( 1 - 6.37T + 43T^{2} \)
47 \( 1 - 0.132T + 47T^{2} \)
53 \( 1 - 6.91T + 53T^{2} \)
59 \( 1 + 0.804T + 59T^{2} \)
61 \( 1 + 4.78T + 61T^{2} \)
67 \( 1 - 4.15T + 67T^{2} \)
71 \( 1 - 5.44T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 + 4.63T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 5.77T + 89T^{2} \)
97 \( 1 - 6.07T + 97T^{2} \)
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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

−7.88396171755958716160659104071, −7.42987008386717887322753318992, −6.40544208148873279210763097845, −5.58817177019965317663597066786, −4.80906664062591250378883457126, −4.01381779192349561126403451399, −3.69188578706498926453465355316, −2.09563829097388886651233020660, −1.26701543585755622062549107725, 0, 1.26701543585755622062549107725, 2.09563829097388886651233020660, 3.69188578706498926453465355316, 4.01381779192349561126403451399, 4.80906664062591250378883457126, 5.58817177019965317663597066786, 6.40544208148873279210763097845, 7.42987008386717887322753318992, 7.88396171755958716160659104071

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