Properties

Label 2-5225-1.1-c1-0-105
Degree $2$
Conductor $5225$
Sign $-1$
Analytic cond. $41.7218$
Root an. cond. $6.45924$
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  − 2.61·2-s − 2.17·3-s + 4.85·4-s + 5.70·6-s − 4.94·7-s − 7.47·8-s + 1.74·9-s + 11-s − 10.5·12-s + 4.20·13-s + 12.9·14-s + 9.84·16-s + 6.38·17-s − 4.56·18-s + 19-s + 10.7·21-s − 2.61·22-s − 6.66·23-s + 16.2·24-s − 11.0·26-s + 2.73·27-s − 24.0·28-s − 6.65·29-s − 9.53·31-s − 10.8·32-s − 2.17·33-s − 16.7·34-s + ⋯
L(s)  = 1  − 1.85·2-s − 1.25·3-s + 2.42·4-s + 2.32·6-s − 1.87·7-s − 2.64·8-s + 0.581·9-s + 0.301·11-s − 3.05·12-s + 1.16·13-s + 3.46·14-s + 2.46·16-s + 1.54·17-s − 1.07·18-s + 0.229·19-s + 2.35·21-s − 0.558·22-s − 1.38·23-s + 3.32·24-s − 2.15·26-s + 0.526·27-s − 4.53·28-s − 1.23·29-s − 1.71·31-s − 1.91·32-s − 0.379·33-s − 2.86·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(41.7218\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 + 2.17T + 3T^{2} \)
7 \( 1 + 4.94T + 7T^{2} \)
13 \( 1 - 4.20T + 13T^{2} \)
17 \( 1 - 6.38T + 17T^{2} \)
23 \( 1 + 6.66T + 23T^{2} \)
29 \( 1 + 6.65T + 29T^{2} \)
31 \( 1 + 9.53T + 31T^{2} \)
37 \( 1 - 3.69T + 37T^{2} \)
41 \( 1 + 1.08T + 41T^{2} \)
43 \( 1 + 9.58T + 43T^{2} \)
47 \( 1 + 2.90T + 47T^{2} \)
53 \( 1 - 6.75T + 53T^{2} \)
59 \( 1 + 0.613T + 59T^{2} \)
61 \( 1 - 1.33T + 61T^{2} \)
67 \( 1 + 0.386T + 67T^{2} \)
71 \( 1 - 9.38T + 71T^{2} \)
73 \( 1 - 1.97T + 73T^{2} \)
79 \( 1 + 6.61T + 79T^{2} \)
83 \( 1 - 0.100T + 83T^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−7.86845595177778095568443856029, −7.10364852013907742644928446902, −6.54397911164910886239194336259, −5.87343615215718459755776194521, −5.62476340948502968407864271366, −3.73649374790581478312626810831, −3.22420461124456401758565875172, −1.86104071881620734520702442742, −0.820824127909821225253441687644, 0, 0.820824127909821225253441687644, 1.86104071881620734520702442742, 3.22420461124456401758565875172, 3.73649374790581478312626810831, 5.62476340948502968407864271366, 5.87343615215718459755776194521, 6.54397911164910886239194336259, 7.10364852013907742644928446902, 7.86845595177778095568443856029

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