Properties

Label 2-6025-1.1-c1-0-151
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  − 1.19·2-s − 1.42·3-s − 0.566·4-s + 1.70·6-s + 1.72·7-s + 3.07·8-s − 0.973·9-s − 6.17·11-s + 0.806·12-s − 2.79·13-s − 2.06·14-s − 2.54·16-s − 1.90·17-s + 1.16·18-s + 1.36·19-s − 2.45·21-s + 7.39·22-s − 1.31·23-s − 4.37·24-s + 3.34·26-s + 5.65·27-s − 0.977·28-s + 6.68·29-s + 9.60·31-s − 3.09·32-s + 8.78·33-s + 2.27·34-s + ⋯
L(s)  = 1  − 0.846·2-s − 0.821·3-s − 0.283·4-s + 0.695·6-s + 0.652·7-s + 1.08·8-s − 0.324·9-s − 1.86·11-s + 0.232·12-s − 0.775·13-s − 0.552·14-s − 0.636·16-s − 0.461·17-s + 0.274·18-s + 0.313·19-s − 0.535·21-s + 1.57·22-s − 0.274·23-s − 0.892·24-s + 0.656·26-s + 1.08·27-s − 0.184·28-s + 1.24·29-s + 1.72·31-s − 0.547·32-s + 1.52·33-s + 0.390·34-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 \( 1 + 1.19T + 2T^{2} \)
3 \( 1 + 1.42T + 3T^{2} \)
7 \( 1 - 1.72T + 7T^{2} \)
11 \( 1 + 6.17T + 11T^{2} \)
13 \( 1 + 2.79T + 13T^{2} \)
17 \( 1 + 1.90T + 17T^{2} \)
19 \( 1 - 1.36T + 19T^{2} \)
23 \( 1 + 1.31T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 9.60T + 31T^{2} \)
37 \( 1 + 2.69T + 37T^{2} \)
41 \( 1 - 0.155T + 41T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 - 0.0487T + 47T^{2} \)
53 \( 1 - 6.75T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 8.73T + 61T^{2} \)
67 \( 1 - 2.97T + 67T^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 0.436T + 83T^{2} \)
89 \( 1 - 5.40T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−7.945519500951555117659418325457, −7.21987004784238398694623293111, −6.34194042247590478481484284899, −5.38269234028942744093128052883, −4.93100580398900612775315590311, −4.46024808789393961003141216944, −2.96372637209096629529811230626, −2.19218843044338781419420591098, −0.876386315022375924962971823603, 0, 0.876386315022375924962971823603, 2.19218843044338781419420591098, 2.96372637209096629529811230626, 4.46024808789393961003141216944, 4.93100580398900612775315590311, 5.38269234028942744093128052883, 6.34194042247590478481484284899, 7.21987004784238398694623293111, 7.945519500951555117659418325457

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