Properties

Label 2-5225-1.1-c1-0-124
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  − 1.65·2-s − 3.32·3-s + 0.743·4-s + 5.50·6-s − 2.81·7-s + 2.08·8-s + 8.06·9-s − 11-s − 2.47·12-s + 5.98·13-s + 4.65·14-s − 4.93·16-s + 6.49·17-s − 13.3·18-s − 19-s + 9.35·21-s + 1.65·22-s − 2.77·23-s − 6.92·24-s − 9.90·26-s − 16.8·27-s − 2.09·28-s − 1.78·29-s − 3.15·31-s + 4.01·32-s + 3.32·33-s − 10.7·34-s + ⋯
L(s)  = 1  − 1.17·2-s − 1.92·3-s + 0.371·4-s + 2.24·6-s − 1.06·7-s + 0.735·8-s + 2.68·9-s − 0.301·11-s − 0.713·12-s + 1.65·13-s + 1.24·14-s − 1.23·16-s + 1.57·17-s − 3.14·18-s − 0.229·19-s + 2.04·21-s + 0.353·22-s − 0.577·23-s − 1.41·24-s − 1.94·26-s − 3.24·27-s − 0.395·28-s − 0.331·29-s − 0.567·31-s + 0.708·32-s + 0.579·33-s − 1.84·34-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 \( 1 + 1.65T + 2T^{2} \)
3 \( 1 + 3.32T + 3T^{2} \)
7 \( 1 + 2.81T + 7T^{2} \)
13 \( 1 - 5.98T + 13T^{2} \)
17 \( 1 - 6.49T + 17T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + 3.15T + 31T^{2} \)
37 \( 1 + 3.90T + 37T^{2} \)
41 \( 1 + 4.63T + 41T^{2} \)
43 \( 1 - 2.40T + 43T^{2} \)
47 \( 1 - 4.10T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 5.33T + 59T^{2} \)
61 \( 1 - 7.18T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 0.437T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 4.44T + 79T^{2} \)
83 \( 1 + 17.7T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + 4.99T + 97T^{2} \)
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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

−7.81611462424291358625211536353, −7.04400783224630035450988090521, −6.45846037344967725469358597337, −5.77587232729119860694577277506, −5.26250592461083963968837493915, −4.15189801884159778289470515593, −3.47796297777766996595124710141, −1.68360697737682742178596023455, −0.908605004971853356444654691527, 0, 0.908605004971853356444654691527, 1.68360697737682742178596023455, 3.47796297777766996595124710141, 4.15189801884159778289470515593, 5.26250592461083963968837493915, 5.77587232729119860694577277506, 6.45846037344967725469358597337, 7.04400783224630035450988090521, 7.81611462424291358625211536353

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