Properties

Label 2-5225-1.1-c1-0-117
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  − 2.47·2-s − 3.11·3-s + 4.12·4-s + 7.70·6-s − 0.568·7-s − 5.24·8-s + 6.71·9-s − 11-s − 12.8·12-s − 1.23·13-s + 1.40·14-s + 4.74·16-s − 1.06·17-s − 16.6·18-s − 19-s + 1.77·21-s + 2.47·22-s + 5.72·23-s + 16.3·24-s + 3.05·26-s − 11.5·27-s − 2.34·28-s + 1.16·29-s − 0.252·31-s − 1.23·32-s + 3.11·33-s + 2.62·34-s + ⋯
L(s)  = 1  − 1.74·2-s − 1.79·3-s + 2.06·4-s + 3.14·6-s − 0.214·7-s − 1.85·8-s + 2.23·9-s − 0.301·11-s − 3.70·12-s − 0.342·13-s + 0.376·14-s + 1.18·16-s − 0.257·17-s − 3.91·18-s − 0.229·19-s + 0.386·21-s + 0.527·22-s + 1.19·23-s + 3.33·24-s + 0.599·26-s − 2.22·27-s − 0.442·28-s + 0.215·29-s − 0.0453·31-s − 0.218·32-s + 0.542·33-s + 0.450·34-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 \( 1 + 2.47T + 2T^{2} \)
3 \( 1 + 3.11T + 3T^{2} \)
7 \( 1 + 0.568T + 7T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 + 1.06T + 17T^{2} \)
23 \( 1 - 5.72T + 23T^{2} \)
29 \( 1 - 1.16T + 29T^{2} \)
31 \( 1 + 0.252T + 31T^{2} \)
37 \( 1 - 0.974T + 37T^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 - 3.31T + 43T^{2} \)
47 \( 1 - 2.74T + 47T^{2} \)
53 \( 1 + 4.08T + 53T^{2} \)
59 \( 1 + 2.54T + 59T^{2} \)
61 \( 1 - 7.49T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 1.42T + 71T^{2} \)
73 \( 1 + 4.15T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 6.03T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−7.71611686743109248389066337849, −7.16593202301214695708780710719, −6.57745632664798273395200787570, −5.99047459301746900431266524913, −5.13284266290911815785267248358, −4.38955242840296990446789570928, −2.94360109217560036850298157038, −1.78668489492458913113260871263, −0.863247103816250627771222849175, 0, 0.863247103816250627771222849175, 1.78668489492458913113260871263, 2.94360109217560036850298157038, 4.38955242840296990446789570928, 5.13284266290911815785267248358, 5.99047459301746900431266524913, 6.57745632664798273395200787570, 7.16593202301214695708780710719, 7.71611686743109248389066337849

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