Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.30·2-s − 1.88·3-s − 0.287·4-s + 2.46·6-s + 1.28·7-s + 2.99·8-s + 0.550·9-s + 0.698·11-s + 0.541·12-s − 5.53·13-s − 1.68·14-s − 3.34·16-s + 3.43·17-s − 0.720·18-s − 3.98·19-s − 2.42·21-s − 0.914·22-s + 6.27·23-s − 5.64·24-s + 7.24·26-s + 4.61·27-s − 0.369·28-s + 6.15·29-s − 0.212·31-s − 1.61·32-s − 1.31·33-s − 4.49·34-s + ⋯
L(s)  = 1  − 0.925·2-s − 1.08·3-s − 0.143·4-s + 1.00·6-s + 0.486·7-s + 1.05·8-s + 0.183·9-s + 0.210·11-s + 0.156·12-s − 1.53·13-s − 0.450·14-s − 0.835·16-s + 0.832·17-s − 0.169·18-s − 0.914·19-s − 0.529·21-s − 0.195·22-s + 1.30·23-s − 1.15·24-s + 1.42·26-s + 0.888·27-s − 0.0698·28-s + 1.14·29-s − 0.0381·31-s − 0.284·32-s − 0.229·33-s − 0.770·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 1.30T + 2T^{2} \)
3 \( 1 + 1.88T + 3T^{2} \)
7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 - 0.698T + 11T^{2} \)
13 \( 1 + 5.53T + 13T^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 + 3.98T + 19T^{2} \)
23 \( 1 - 6.27T + 23T^{2} \)
29 \( 1 - 6.15T + 29T^{2} \)
31 \( 1 + 0.212T + 31T^{2} \)
37 \( 1 + 5.91T + 37T^{2} \)
41 \( 1 + 9.65T + 41T^{2} \)
43 \( 1 + 7.78T + 43T^{2} \)
47 \( 1 - 3.86T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 7.38T + 67T^{2} \)
71 \( 1 + 6.00T + 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 + 6.80T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 - 1.74T + 89T^{2} \)
97 \( 1 - 0.714T + 97T^{2} \)
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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

−7.79551109451281616116857541655, −7.00122894964373722652844936373, −6.56103786275218902083376569004, −5.28536340655025904300428686251, −5.07584419706933863171416824840, −4.36789249536902137071058087046, −3.12530584994224424905513951226, −1.95590663826772561839969901111, −0.941584792792833557930637510237, 0, 0.941584792792833557930637510237, 1.95590663826772561839969901111, 3.12530584994224424905513951226, 4.36789249536902137071058087046, 5.07584419706933863171416824840, 5.28536340655025904300428686251, 6.56103786275218902083376569004, 7.00122894964373722652844936373, 7.79551109451281616116857541655

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