Properties

Label 2-6025-1.1-c1-0-228
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.06·2-s + 2.17·3-s + 2.27·4-s − 4.49·6-s + 2.39·7-s − 0.576·8-s + 1.71·9-s + 4.70·11-s + 4.94·12-s − 0.253·13-s − 4.94·14-s − 3.36·16-s + 7.11·17-s − 3.54·18-s + 7.42·19-s + 5.19·21-s − 9.72·22-s − 0.555·23-s − 1.25·24-s + 0.524·26-s − 2.79·27-s + 5.44·28-s − 0.396·29-s + 9.01·31-s + 8.11·32-s + 10.2·33-s − 14.7·34-s + ⋯
L(s)  = 1  − 1.46·2-s + 1.25·3-s + 1.13·4-s − 1.83·6-s + 0.903·7-s − 0.203·8-s + 0.571·9-s + 1.41·11-s + 1.42·12-s − 0.0703·13-s − 1.32·14-s − 0.841·16-s + 1.72·17-s − 0.836·18-s + 1.70·19-s + 1.13·21-s − 2.07·22-s − 0.115·23-s − 0.255·24-s + 0.102·26-s − 0.536·27-s + 1.02·28-s − 0.0735·29-s + 1.61·31-s + 1.43·32-s + 1.77·33-s − 2.52·34-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: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207944516\)
\(L(\frac12)\) \(\approx\) \(2.207944516\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.06T + 2T^{2} \)
3 \( 1 - 2.17T + 3T^{2} \)
7 \( 1 - 2.39T + 7T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 + 0.253T + 13T^{2} \)
17 \( 1 - 7.11T + 17T^{2} \)
19 \( 1 - 7.42T + 19T^{2} \)
23 \( 1 + 0.555T + 23T^{2} \)
29 \( 1 + 0.396T + 29T^{2} \)
31 \( 1 - 9.01T + 31T^{2} \)
37 \( 1 - 2.41T + 37T^{2} \)
41 \( 1 + 1.30T + 41T^{2} \)
43 \( 1 + 1.26T + 43T^{2} \)
47 \( 1 - 3.17T + 47T^{2} \)
53 \( 1 - 6.62T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 5.87T + 61T^{2} \)
67 \( 1 + 4.45T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 - 2.46T + 73T^{2} \)
79 \( 1 + 2.95T + 79T^{2} \)
83 \( 1 - 7.18T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−8.228533456663777545911783892976, −7.62277773432817524851213079012, −7.24837988404101986296985500416, −6.20629913851096966426111331400, −5.21523426899566041356152239377, −4.23953667437241661284395906443, −3.38203348931974067262671417422, −2.59589910556346969379938303340, −1.44738315055641810546461760038, −1.11034755874026456908042260267, 1.11034755874026456908042260267, 1.44738315055641810546461760038, 2.59589910556346969379938303340, 3.38203348931974067262671417422, 4.23953667437241661284395906443, 5.21523426899566041356152239377, 6.20629913851096966426111331400, 7.24837988404101986296985500416, 7.62277773432817524851213079012, 8.228533456663777545911783892976

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