Properties

Label 2-6025-1.1-c1-0-338
Degree $2$
Conductor $6025$
Sign $-1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
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  + 1.92·2-s − 0.557·3-s + 1.69·4-s − 1.07·6-s + 0.976·7-s − 0.583·8-s − 2.68·9-s + 5.68·11-s − 0.945·12-s + 0.519·13-s + 1.87·14-s − 4.51·16-s − 2.78·17-s − 5.17·18-s − 2.82·19-s − 0.544·21-s + 10.9·22-s − 7.25·23-s + 0.325·24-s + 0.997·26-s + 3.17·27-s + 1.65·28-s − 2.98·29-s − 4.59·31-s − 7.51·32-s − 3.16·33-s − 5.36·34-s + ⋯
L(s)  = 1  + 1.35·2-s − 0.321·3-s + 0.848·4-s − 0.437·6-s + 0.369·7-s − 0.206·8-s − 0.896·9-s + 1.71·11-s − 0.273·12-s + 0.143·13-s + 0.501·14-s − 1.12·16-s − 0.676·17-s − 1.21·18-s − 0.647·19-s − 0.118·21-s + 2.32·22-s − 1.51·23-s + 0.0664·24-s + 0.195·26-s + 0.610·27-s + 0.313·28-s − 0.554·29-s − 0.825·31-s − 1.32·32-s − 0.551·33-s − 0.919·34-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: $-1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
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 \( 1 - 1.92T + 2T^{2} \)
3 \( 1 + 0.557T + 3T^{2} \)
7 \( 1 - 0.976T + 7T^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
13 \( 1 - 0.519T + 13T^{2} \)
17 \( 1 + 2.78T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 7.25T + 23T^{2} \)
29 \( 1 + 2.98T + 29T^{2} \)
31 \( 1 + 4.59T + 31T^{2} \)
37 \( 1 - 9.40T + 37T^{2} \)
41 \( 1 - 1.99T + 41T^{2} \)
43 \( 1 - 5.68T + 43T^{2} \)
47 \( 1 + 1.33T + 47T^{2} \)
53 \( 1 + 2.40T + 53T^{2} \)
59 \( 1 - 5.64T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 5.66T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 1.00T + 97T^{2} \)
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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

−7.52582020373220891174229879870, −6.61584488786481084427679309466, −5.97325203468116265901507461051, −5.81097822935669983195976795020, −4.56256239459255595075792842092, −4.25898556264701261192404146539, −3.49689549532145624754431709611, −2.54175904666301384764307699547, −1.60149986900531694957543430074, 0, 1.60149986900531694957543430074, 2.54175904666301384764307699547, 3.49689549532145624754431709611, 4.25898556264701261192404146539, 4.56256239459255595075792842092, 5.81097822935669983195976795020, 5.97325203468116265901507461051, 6.61584488786481084427679309466, 7.52582020373220891174229879870

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