Properties

Label 2-6025-1.1-c1-0-16
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  − 0.193·2-s − 2.50·3-s − 1.96·4-s + 0.483·6-s + 0.166·7-s + 0.766·8-s + 3.25·9-s − 0.120·11-s + 4.90·12-s − 4.23·13-s − 0.0320·14-s + 3.77·16-s − 5.01·17-s − 0.629·18-s − 2.77·19-s − 0.415·21-s + 0.0233·22-s − 5.34·23-s − 1.91·24-s + 0.819·26-s − 0.645·27-s − 0.325·28-s + 0.290·29-s + 3.57·31-s − 2.26·32-s + 0.301·33-s + 0.970·34-s + ⋯
L(s)  = 1  − 0.136·2-s − 1.44·3-s − 0.981·4-s + 0.197·6-s + 0.0627·7-s + 0.270·8-s + 1.08·9-s − 0.0363·11-s + 1.41·12-s − 1.17·13-s − 0.00857·14-s + 0.944·16-s − 1.21·17-s − 0.148·18-s − 0.637·19-s − 0.0906·21-s + 0.00496·22-s − 1.11·23-s − 0.391·24-s + 0.160·26-s − 0.124·27-s − 0.0615·28-s + 0.0538·29-s + 0.641·31-s − 0.399·32-s + 0.0524·33-s + 0.166·34-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

\(L(1)\) \(\approx\) \(0.1305424587\)
\(L(\frac12)\) \(\approx\) \(0.1305424587\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.193T + 2T^{2} \)
3 \( 1 + 2.50T + 3T^{2} \)
7 \( 1 - 0.166T + 7T^{2} \)
11 \( 1 + 0.120T + 11T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 + 5.01T + 17T^{2} \)
19 \( 1 + 2.77T + 19T^{2} \)
23 \( 1 + 5.34T + 23T^{2} \)
29 \( 1 - 0.290T + 29T^{2} \)
31 \( 1 - 3.57T + 31T^{2} \)
37 \( 1 + 0.104T + 37T^{2} \)
41 \( 1 - 7.23T + 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 + 1.90T + 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 + 6.34T + 59T^{2} \)
61 \( 1 + 0.0566T + 61T^{2} \)
67 \( 1 + 6.47T + 67T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 4.84T + 83T^{2} \)
89 \( 1 + 5.92T + 89T^{2} \)
97 \( 1 - 4.77T + 97T^{2} \)
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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

−8.082481163509190902275421092553, −7.31265547647737000073013401740, −6.47235050128768789166164858955, −5.94477508735196408552280784869, −5.10224800072260849315662237913, −4.59067728854369615504110622106, −4.09354139226530300077860541284, −2.72674747529396413553153869314, −1.54667391257635251053096928635, −0.21330483999755900099387615792, 0.21330483999755900099387615792, 1.54667391257635251053096928635, 2.72674747529396413553153869314, 4.09354139226530300077860541284, 4.59067728854369615504110622106, 5.10224800072260849315662237913, 5.94477508735196408552280784869, 6.47235050128768789166164858955, 7.31265547647737000073013401740, 8.082481163509190902275421092553

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