Properties

Label 2-5225-1.1-c1-0-136
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  − 1.38·2-s + 0.179·3-s − 0.0852·4-s − 0.248·6-s − 4.32·7-s + 2.88·8-s − 2.96·9-s + 11-s − 0.0153·12-s − 0.663·13-s + 5.98·14-s − 3.82·16-s + 5.90·17-s + 4.10·18-s − 19-s − 0.776·21-s − 1.38·22-s − 0.697·23-s + 0.518·24-s + 0.917·26-s − 1.07·27-s + 0.368·28-s − 3.32·29-s + 10.4·31-s − 0.481·32-s + 0.179·33-s − 8.16·34-s + ⋯
L(s)  = 1  − 0.978·2-s + 0.103·3-s − 0.0426·4-s − 0.101·6-s − 1.63·7-s + 1.02·8-s − 0.989·9-s + 0.301·11-s − 0.00441·12-s − 0.183·13-s + 1.59·14-s − 0.955·16-s + 1.43·17-s + 0.967·18-s − 0.229·19-s − 0.169·21-s − 0.295·22-s − 0.145·23-s + 0.105·24-s + 0.179·26-s − 0.206·27-s + 0.0696·28-s − 0.618·29-s + 1.88·31-s − 0.0851·32-s + 0.0312·33-s − 1.40·34-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 \( 1 + 1.38T + 2T^{2} \)
3 \( 1 - 0.179T + 3T^{2} \)
7 \( 1 + 4.32T + 7T^{2} \)
13 \( 1 + 0.663T + 13T^{2} \)
17 \( 1 - 5.90T + 17T^{2} \)
23 \( 1 + 0.697T + 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 + 2.76T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 8.10T + 53T^{2} \)
59 \( 1 + 0.264T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 + 9.64T + 71T^{2} \)
73 \( 1 - 0.803T + 73T^{2} \)
79 \( 1 - 8.25T + 79T^{2} \)
83 \( 1 + 1.39T + 83T^{2} \)
89 \( 1 - 9.69T + 89T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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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

−8.165584102499505898034264081408, −7.18065956916565783704162422213, −6.59200147548870250567730331932, −5.78702761618414625702087525513, −5.05524068819303125007362104066, −3.84605329139796552742388472911, −3.27537907680748163121417442347, −2.33607034394152551620339432604, −0.965308125644210428234921606095, 0, 0.965308125644210428234921606095, 2.33607034394152551620339432604, 3.27537907680748163121417442347, 3.84605329139796552742388472911, 5.05524068819303125007362104066, 5.78702761618414625702087525513, 6.59200147548870250567730331932, 7.18065956916565783704162422213, 8.165584102499505898034264081408

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