Properties

Label 2-5225-1.1-c1-0-279
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  + 2.04·2-s + 1.09·3-s + 2.16·4-s + 2.24·6-s + 0.469·7-s + 0.346·8-s − 1.79·9-s − 11-s + 2.38·12-s − 2.14·13-s + 0.957·14-s − 3.63·16-s − 4.47·17-s − 3.66·18-s − 19-s + 0.515·21-s − 2.04·22-s − 0.200·23-s + 0.380·24-s − 4.37·26-s − 5.26·27-s + 1.01·28-s − 6.92·29-s + 4.65·31-s − 8.10·32-s − 1.09·33-s − 9.14·34-s + ⋯
L(s)  = 1  + 1.44·2-s + 0.634·3-s + 1.08·4-s + 0.915·6-s + 0.177·7-s + 0.122·8-s − 0.597·9-s − 0.301·11-s + 0.687·12-s − 0.593·13-s + 0.256·14-s − 0.907·16-s − 1.08·17-s − 0.863·18-s − 0.229·19-s + 0.112·21-s − 0.435·22-s − 0.0419·23-s + 0.0777·24-s − 0.857·26-s − 1.01·27-s + 0.192·28-s − 1.28·29-s + 0.836·31-s − 1.43·32-s − 0.191·33-s − 1.56·34-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 \( 1 - 2.04T + 2T^{2} \)
3 \( 1 - 1.09T + 3T^{2} \)
7 \( 1 - 0.469T + 7T^{2} \)
13 \( 1 + 2.14T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
23 \( 1 + 0.200T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 4.65T + 31T^{2} \)
37 \( 1 - 7.22T + 37T^{2} \)
41 \( 1 + 3.34T + 41T^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 - 2.69T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 + 5.04T + 61T^{2} \)
67 \( 1 + 7.47T + 67T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 5.88T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 0.794T + 89T^{2} \)
97 \( 1 + 17.5T + 97T^{2} \)
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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

−7.86429523460475020426967182275, −6.89439040984256807658924949857, −6.27549289134375623029935557498, −5.49245880091069572213540646819, −4.83555279603746813715582997200, −4.15869849638887161412617317454, −3.34234165260477157637872816565, −2.61881748081244937500683330322, −1.99078771617525917834672030086, 0, 1.99078771617525917834672030086, 2.61881748081244937500683330322, 3.34234165260477157637872816565, 4.15869849638887161412617317454, 4.83555279603746813715582997200, 5.49245880091069572213540646819, 6.27549289134375623029935557498, 6.89439040984256807658924949857, 7.86429523460475020426967182275

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