Properties

Label 2-5225-1.1-c1-0-131
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  − 0.386·2-s − 0.261·3-s − 1.85·4-s + 0.101·6-s − 4.13·7-s + 1.48·8-s − 2.93·9-s + 11-s + 0.484·12-s − 0.244·13-s + 1.59·14-s + 3.12·16-s + 4.03·17-s + 1.13·18-s − 19-s + 1.08·21-s − 0.386·22-s − 0.483·23-s − 0.389·24-s + 0.0946·26-s + 1.55·27-s + 7.65·28-s + 5.24·29-s − 9.76·31-s − 4.18·32-s − 0.261·33-s − 1.56·34-s + ⋯
L(s)  = 1  − 0.273·2-s − 0.150·3-s − 0.925·4-s + 0.0412·6-s − 1.56·7-s + 0.526·8-s − 0.977·9-s + 0.301·11-s + 0.139·12-s − 0.0679·13-s + 0.427·14-s + 0.781·16-s + 0.979·17-s + 0.267·18-s − 0.229·19-s + 0.236·21-s − 0.0823·22-s − 0.100·23-s − 0.0794·24-s + 0.0185·26-s + 0.298·27-s + 1.44·28-s + 0.973·29-s − 1.75·31-s − 0.739·32-s − 0.0455·33-s − 0.267·34-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 \( 1 + 0.386T + 2T^{2} \)
3 \( 1 + 0.261T + 3T^{2} \)
7 \( 1 + 4.13T + 7T^{2} \)
13 \( 1 + 0.244T + 13T^{2} \)
17 \( 1 - 4.03T + 17T^{2} \)
23 \( 1 + 0.483T + 23T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + 9.76T + 31T^{2} \)
37 \( 1 - 1.52T + 37T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 - 7.60T + 43T^{2} \)
47 \( 1 - 9.01T + 47T^{2} \)
53 \( 1 + 5.86T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 - 7.58T + 61T^{2} \)
67 \( 1 + 2.72T + 67T^{2} \)
71 \( 1 - 2.89T + 71T^{2} \)
73 \( 1 - 7.24T + 73T^{2} \)
79 \( 1 - 5.91T + 79T^{2} \)
83 \( 1 - 4.94T + 83T^{2} \)
89 \( 1 + 3.41T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.898665654164049850900030709889, −7.22666438880525863595516405055, −6.20746093160059671742200921077, −5.81566257953790786109703152825, −5.00065443767400519112208484846, −3.94880518588701047703359985822, −3.39684867817203973870543458959, −2.53283995690340801960197984403, −0.953514533044979032873544719456, 0, 0.953514533044979032873544719456, 2.53283995690340801960197984403, 3.39684867817203973870543458959, 3.94880518588701047703359985822, 5.00065443767400519112208484846, 5.81566257953790786109703152825, 6.20746093160059671742200921077, 7.22666438880525863595516405055, 7.898665654164049850900030709889

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