Properties

Label 2-5225-1.1-c1-0-116
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  + 1.45·2-s − 0.0791·3-s + 0.106·4-s − 0.114·6-s + 2.96·7-s − 2.74·8-s − 2.99·9-s + 11-s − 0.00845·12-s + 3.94·13-s + 4.29·14-s − 4.20·16-s + 4.79·17-s − 4.34·18-s + 19-s − 0.234·21-s + 1.45·22-s − 5.69·23-s + 0.217·24-s + 5.73·26-s + 0.474·27-s + 0.316·28-s + 2.19·29-s − 8.56·31-s − 0.603·32-s − 0.0791·33-s + 6.95·34-s + ⋯
L(s)  = 1  + 1.02·2-s − 0.0456·3-s + 0.0534·4-s − 0.0468·6-s + 1.11·7-s − 0.971·8-s − 0.997·9-s + 0.301·11-s − 0.00244·12-s + 1.09·13-s + 1.14·14-s − 1.05·16-s + 1.16·17-s − 1.02·18-s + 0.229·19-s − 0.0511·21-s + 0.309·22-s − 1.18·23-s + 0.0443·24-s + 1.12·26-s + 0.0912·27-s + 0.0598·28-s + 0.407·29-s − 1.53·31-s − 0.106·32-s − 0.0137·33-s + 1.19·34-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

\(L(1)\) \(\approx\) \(3.231278452\)
\(L(\frac12)\) \(\approx\) \(3.231278452\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.45T + 2T^{2} \)
3 \( 1 + 0.0791T + 3T^{2} \)
7 \( 1 - 2.96T + 7T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 + 8.56T + 31T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 3.44T + 47T^{2} \)
53 \( 1 + 2.73T + 53T^{2} \)
59 \( 1 + 1.87T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 + 3.73T + 73T^{2} \)
79 \( 1 - 16.6T + 79T^{2} \)
83 \( 1 - 9.90T + 83T^{2} \)
89 \( 1 + 1.30T + 89T^{2} \)
97 \( 1 - 9.85T + 97T^{2} \)
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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

−7.953132351056191832346691657048, −7.76285104068151950574513244374, −6.33357945955450994265406453837, −5.84266899677852245244258972938, −5.37342937701188278994121313934, −4.48590803895728360653786336032, −3.82435238188862366732566400136, −3.11683241127994745607833223864, −2.07706574986226365313053845228, −0.856612234651718248626772101824, 0.856612234651718248626772101824, 2.07706574986226365313053845228, 3.11683241127994745607833223864, 3.82435238188862366732566400136, 4.48590803895728360653786336032, 5.37342937701188278994121313934, 5.84266899677852245244258972938, 6.33357945955450994265406453837, 7.76285104068151950574513244374, 7.953132351056191832346691657048

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