Properties

Label 2-5225-1.1-c1-0-129
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  + 1.24·2-s − 3.09·3-s − 0.453·4-s − 3.84·6-s − 3.83·7-s − 3.05·8-s + 6.57·9-s + 11-s + 1.40·12-s + 2.44·13-s − 4.77·14-s − 2.88·16-s − 4.44·17-s + 8.17·18-s + 19-s + 11.8·21-s + 1.24·22-s + 5.34·23-s + 9.44·24-s + 3.04·26-s − 11.0·27-s + 1.73·28-s − 9.42·29-s − 2.25·31-s + 2.50·32-s − 3.09·33-s − 5.53·34-s + ⋯
L(s)  = 1  + 0.879·2-s − 1.78·3-s − 0.226·4-s − 1.57·6-s − 1.45·7-s − 1.07·8-s + 2.19·9-s + 0.301·11-s + 0.404·12-s + 0.678·13-s − 1.27·14-s − 0.722·16-s − 1.07·17-s + 1.92·18-s + 0.229·19-s + 2.59·21-s + 0.265·22-s + 1.11·23-s + 1.92·24-s + 0.596·26-s − 2.12·27-s + 0.328·28-s − 1.75·29-s − 0.404·31-s + 0.443·32-s − 0.538·33-s − 0.948·34-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 \( 1 - 1.24T + 2T^{2} \)
3 \( 1 + 3.09T + 3T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
23 \( 1 - 5.34T + 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + 2.25T + 31T^{2} \)
37 \( 1 - 9.50T + 37T^{2} \)
41 \( 1 - 6.98T + 41T^{2} \)
43 \( 1 + 3.25T + 43T^{2} \)
47 \( 1 - 4.99T + 47T^{2} \)
53 \( 1 - 6.72T + 53T^{2} \)
59 \( 1 - 3.81T + 59T^{2} \)
61 \( 1 - 7.33T + 61T^{2} \)
67 \( 1 - 8.95T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 - 3.88T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 8.59T + 83T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 - 8.42T + 97T^{2} \)
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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

−7.38325086121399893294923969870, −6.68647956820758294171495845826, −6.24827534321892340261537339494, −5.64821207701919989181533892363, −5.11368206976495491915989055081, −4.08239141370581576638327156997, −3.79820242955941902927185141782, −2.59035317812467183445759848165, −0.941122099269063823711284613491, 0, 0.941122099269063823711284613491, 2.59035317812467183445759848165, 3.79820242955941902927185141782, 4.08239141370581576638327156997, 5.11368206976495491915989055081, 5.64821207701919989181533892363, 6.24827534321892340261537339494, 6.68647956820758294171495845826, 7.38325086121399893294923969870

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