Properties

Label 2-6025-1.1-c1-0-147
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  − 0.355·2-s − 0.886·3-s − 1.87·4-s + 0.315·6-s + 4.33·7-s + 1.37·8-s − 2.21·9-s + 4.19·11-s + 1.66·12-s − 1.37·13-s − 1.54·14-s + 3.25·16-s + 2.36·17-s + 0.787·18-s + 8.50·19-s − 3.84·21-s − 1.49·22-s − 3.14·23-s − 1.22·24-s + 0.489·26-s + 4.62·27-s − 8.12·28-s + 10.6·29-s + 1.37·31-s − 3.91·32-s − 3.72·33-s − 0.841·34-s + ⋯
L(s)  = 1  − 0.251·2-s − 0.511·3-s − 0.936·4-s + 0.128·6-s + 1.63·7-s + 0.487·8-s − 0.738·9-s + 1.26·11-s + 0.479·12-s − 0.381·13-s − 0.412·14-s + 0.814·16-s + 0.573·17-s + 0.185·18-s + 1.95·19-s − 0.838·21-s − 0.318·22-s − 0.655·23-s − 0.249·24-s + 0.0960·26-s + 0.889·27-s − 1.53·28-s + 1.97·29-s + 0.247·31-s − 0.692·32-s − 0.648·33-s − 0.144·34-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

\(L(1)\) \(\approx\) \(1.666923809\)
\(L(\frac12)\) \(\approx\) \(1.666923809\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.355T + 2T^{2} \)
3 \( 1 + 0.886T + 3T^{2} \)
7 \( 1 - 4.33T + 7T^{2} \)
11 \( 1 - 4.19T + 11T^{2} \)
13 \( 1 + 1.37T + 13T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 - 8.50T + 19T^{2} \)
23 \( 1 + 3.14T + 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 1.37T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + 2.36T + 41T^{2} \)
43 \( 1 - 2.46T + 43T^{2} \)
47 \( 1 - 4.36T + 47T^{2} \)
53 \( 1 + 8.75T + 53T^{2} \)
59 \( 1 - 8.02T + 59T^{2} \)
61 \( 1 - 3.25T + 61T^{2} \)
67 \( 1 - 5.05T + 67T^{2} \)
71 \( 1 + 7.73T + 71T^{2} \)
73 \( 1 + 1.97T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−8.089655432170881181966589297544, −7.64545186273659711582099177945, −6.65629634782759644070940195216, −5.70694760283698344365114351918, −5.16616754426069836868562115747, −4.62156800242752634668215811750, −3.84317123503766496656671549516, −2.79705936641622199676710365140, −1.39211496411858464562559026068, −0.859720238510854618348255518005, 0.859720238510854618348255518005, 1.39211496411858464562559026068, 2.79705936641622199676710365140, 3.84317123503766496656671549516, 4.62156800242752634668215811750, 5.16616754426069836868562115747, 5.70694760283698344365114351918, 6.65629634782759644070940195216, 7.64545186273659711582099177945, 8.089655432170881181966589297544

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