Properties

Label 2-6025-1.1-c1-0-319
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  + 2.18·2-s − 2.05·3-s + 2.76·4-s − 4.47·6-s + 1.39·7-s + 1.66·8-s + 1.20·9-s + 3.94·11-s − 5.66·12-s − 5.07·13-s + 3.03·14-s − 1.89·16-s − 0.309·17-s + 2.63·18-s − 1.55·19-s − 2.85·21-s + 8.60·22-s + 0.977·23-s − 3.41·24-s − 11.0·26-s + 3.67·27-s + 3.84·28-s − 0.431·29-s + 5.32·31-s − 7.46·32-s − 8.08·33-s − 0.675·34-s + ⋯
L(s)  = 1  + 1.54·2-s − 1.18·3-s + 1.38·4-s − 1.82·6-s + 0.526·7-s + 0.588·8-s + 0.402·9-s + 1.18·11-s − 1.63·12-s − 1.40·13-s + 0.812·14-s − 0.473·16-s − 0.0751·17-s + 0.621·18-s − 0.356·19-s − 0.623·21-s + 1.83·22-s + 0.203·23-s − 0.696·24-s − 2.17·26-s + 0.707·27-s + 0.727·28-s − 0.0802·29-s + 0.956·31-s − 1.31·32-s − 1.40·33-s − 0.115·34-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 \( 1 - 2.18T + 2T^{2} \)
3 \( 1 + 2.05T + 3T^{2} \)
7 \( 1 - 1.39T + 7T^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 + 0.309T + 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 - 0.977T + 23T^{2} \)
29 \( 1 + 0.431T + 29T^{2} \)
31 \( 1 - 5.32T + 31T^{2} \)
37 \( 1 + 8.05T + 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 + 6.84T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 1.13T + 67T^{2} \)
71 \( 1 + 0.392T + 71T^{2} \)
73 \( 1 + 1.73T + 73T^{2} \)
79 \( 1 - 2.78T + 79T^{2} \)
83 \( 1 - 6.88T + 83T^{2} \)
89 \( 1 + 3.18T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−7.21870037276077480562060618356, −6.67834892207414858168865267180, −6.19980717911329307209182576617, −5.31319682925552146571799530334, −4.92942984993605060086957826088, −4.36333695105262310511614637096, −3.48946824587914940471724236911, −2.53139608858003393306451301136, −1.49827939623724218936101743275, 0, 1.49827939623724218936101743275, 2.53139608858003393306451301136, 3.48946824587914940471724236911, 4.36333695105262310511614637096, 4.92942984993605060086957826088, 5.31319682925552146571799530334, 6.19980717911329307209182576617, 6.67834892207414858168865267180, 7.21870037276077480562060618356

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