Properties

Label 2-5225-1.1-c1-0-130
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.89·2-s − 3.43·3-s + 1.58·4-s − 6.51·6-s + 4.54·7-s − 0.778·8-s + 8.82·9-s + 11-s − 5.46·12-s + 3.12·13-s + 8.60·14-s − 4.65·16-s + 4.24·17-s + 16.7·18-s + 19-s − 15.6·21-s + 1.89·22-s + 7.93·23-s + 2.67·24-s + 5.91·26-s − 20.0·27-s + 7.21·28-s + 0.981·29-s + 5.23·31-s − 7.25·32-s − 3.43·33-s + 8.04·34-s + ⋯
L(s)  = 1  + 1.33·2-s − 1.98·3-s + 0.794·4-s − 2.66·6-s + 1.71·7-s − 0.275·8-s + 2.94·9-s + 0.301·11-s − 1.57·12-s + 0.865·13-s + 2.29·14-s − 1.16·16-s + 1.03·17-s + 3.94·18-s + 0.229·19-s − 3.40·21-s + 0.403·22-s + 1.65·23-s + 0.546·24-s + 1.15·26-s − 3.85·27-s + 1.36·28-s + 0.182·29-s + 0.940·31-s − 1.28·32-s − 0.598·33-s + 1.38·34-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: \(0\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.951391059\)
\(L(\frac12)\) \(\approx\) \(2.951391059\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.89T + 2T^{2} \)
3 \( 1 + 3.43T + 3T^{2} \)
7 \( 1 - 4.54T + 7T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
23 \( 1 - 7.93T + 23T^{2} \)
29 \( 1 - 0.981T + 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 - 0.410T + 37T^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 + 3.20T + 43T^{2} \)
47 \( 1 + 2.27T + 47T^{2} \)
53 \( 1 + 9.47T + 53T^{2} \)
59 \( 1 + 5.23T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + 2.99T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 8.61T + 89T^{2} \)
97 \( 1 + 6.59T + 97T^{2} \)
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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

−7.86196330961411207484807499658, −7.17614663544911074467494623436, −6.34138313029646554758718172953, −5.84408757807284018855881238498, −5.16861181172217923034127119774, −4.72866273912854546379782721824, −4.24716014731002472222784101112, −3.19598651054950569522782563713, −1.61109632673347589547063695012, −0.951419807506579127262245606042, 0.951419807506579127262245606042, 1.61109632673347589547063695012, 3.19598651054950569522782563713, 4.24716014731002472222784101112, 4.72866273912854546379782721824, 5.16861181172217923034127119774, 5.84408757807284018855881238498, 6.34138313029646554758718172953, 7.17614663544911074467494623436, 7.86196330961411207484807499658

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