Properties

Label 2-5225-1.1-c1-0-101
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  + 2.41·2-s − 2.46·3-s + 3.84·4-s − 5.95·6-s + 1.10·7-s + 4.46·8-s + 3.05·9-s − 11-s − 9.46·12-s − 5.07·13-s + 2.67·14-s + 3.10·16-s + 2.37·17-s + 7.39·18-s − 19-s − 2.72·21-s − 2.41·22-s + 2.73·23-s − 10.9·24-s − 12.2·26-s − 0.142·27-s + 4.25·28-s + 9.42·29-s − 7.43·31-s − 1.42·32-s + 2.46·33-s + 5.75·34-s + ⋯
L(s)  = 1  + 1.70·2-s − 1.42·3-s + 1.92·4-s − 2.42·6-s + 0.418·7-s + 1.57·8-s + 1.01·9-s − 0.301·11-s − 2.73·12-s − 1.40·13-s + 0.715·14-s + 0.775·16-s + 0.577·17-s + 1.74·18-s − 0.229·19-s − 0.594·21-s − 0.515·22-s + 0.570·23-s − 2.24·24-s − 2.40·26-s − 0.0274·27-s + 0.804·28-s + 1.75·29-s − 1.33·31-s − 0.252·32-s + 0.428·33-s + 0.986·34-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

\(L(1)\) \(\approx\) \(3.249799747\)
\(L(\frac12)\) \(\approx\) \(3.249799747\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
3 \( 1 + 2.46T + 3T^{2} \)
7 \( 1 - 1.10T + 7T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
23 \( 1 - 2.73T + 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + 7.43T + 31T^{2} \)
37 \( 1 - 7.39T + 37T^{2} \)
41 \( 1 - 4.35T + 41T^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 - 6.64T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 3.23T + 73T^{2} \)
79 \( 1 + 0.514T + 79T^{2} \)
83 \( 1 - 6.35T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 8.43T + 97T^{2} \)
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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

−7.59060631212962898214672272839, −7.30490273223857889843968611017, −6.32302925386116126604945281877, −5.89212985744717324248102374046, −5.13021210890995079110106888002, −4.78214953016353273986726074696, −4.12480819235227782723262817178, −2.95703368681485128265017947341, −2.22824894865272872818003239316, −0.799285508578500705209909778851, 0.799285508578500705209909778851, 2.22824894865272872818003239316, 2.95703368681485128265017947341, 4.12480819235227782723262817178, 4.78214953016353273986726074696, 5.13021210890995079110106888002, 5.89212985744717324248102374046, 6.32302925386116126604945281877, 7.30490273223857889843968611017, 7.59060631212962898214672272839

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