Properties

Label 2-5225-1.1-c1-0-234
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  + 0.577·2-s + 0.746·3-s − 1.66·4-s + 0.431·6-s + 4.19·7-s − 2.11·8-s − 2.44·9-s − 11-s − 1.24·12-s − 0.324·13-s + 2.42·14-s + 2.10·16-s + 5.00·17-s − 1.41·18-s − 19-s + 3.13·21-s − 0.577·22-s − 8.11·23-s − 1.58·24-s − 0.187·26-s − 4.06·27-s − 6.98·28-s − 3.53·29-s − 3.45·31-s + 5.45·32-s − 0.746·33-s + 2.89·34-s + ⋯
L(s)  = 1  + 0.408·2-s + 0.431·3-s − 0.833·4-s + 0.176·6-s + 1.58·7-s − 0.748·8-s − 0.814·9-s − 0.301·11-s − 0.359·12-s − 0.0899·13-s + 0.647·14-s + 0.527·16-s + 1.21·17-s − 0.332·18-s − 0.229·19-s + 0.683·21-s − 0.123·22-s − 1.69·23-s − 0.322·24-s − 0.0367·26-s − 0.782·27-s − 1.32·28-s − 0.656·29-s − 0.621·31-s + 0.964·32-s − 0.129·33-s + 0.496·34-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 \( 1 - 0.577T + 2T^{2} \)
3 \( 1 - 0.746T + 3T^{2} \)
7 \( 1 - 4.19T + 7T^{2} \)
13 \( 1 + 0.324T + 13T^{2} \)
17 \( 1 - 5.00T + 17T^{2} \)
23 \( 1 + 8.11T + 23T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 + 5.77T + 37T^{2} \)
41 \( 1 - 0.484T + 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + 6.22T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 - 7.16T + 59T^{2} \)
61 \( 1 - 5.33T + 61T^{2} \)
67 \( 1 + 8.20T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 2.88T + 73T^{2} \)
79 \( 1 - 3.95T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 7.47T + 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−7.946631232383100846224297559307, −7.47334149668509205540520263524, −6.10089062239668782283651762870, −5.40992697541744812891551376744, −5.06487367925202319678820731882, −4.05099374802824709313475731003, −3.50751269787072095913438281317, −2.41824787307215002851607943679, −1.49417526420053983440673895278, 0, 1.49417526420053983440673895278, 2.41824787307215002851607943679, 3.50751269787072095913438281317, 4.05099374802824709313475731003, 5.06487367925202319678820731882, 5.40992697541744812891551376744, 6.10089062239668782283651762870, 7.47334149668509205540520263524, 7.946631232383100846224297559307

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