Properties

Label 2-7-1.1-c17-0-5
Degree $2$
Conductor $7$
Sign $1$
Analytic cond. $12.8255$
Root an. cond. $3.58127$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 494.·2-s + 1.66e4·3-s + 1.13e5·4-s + 4.21e5·5-s + 8.25e6·6-s + 5.76e6·7-s − 8.76e6·8-s + 1.49e8·9-s + 2.08e8·10-s − 4.95e8·11-s + 1.89e9·12-s − 1.96e9·13-s + 2.85e9·14-s + 7.03e9·15-s − 1.91e10·16-s + 3.37e10·17-s + 7.40e10·18-s + 1.12e11·19-s + 4.77e10·20-s + 9.62e10·21-s − 2.44e11·22-s − 6.00e11·23-s − 1.46e11·24-s − 5.85e11·25-s − 9.72e11·26-s + 3.43e11·27-s + 6.53e11·28-s + ⋯
L(s)  = 1  + 1.36·2-s + 1.46·3-s + 0.864·4-s + 0.482·5-s + 2.00·6-s + 0.377·7-s − 0.184·8-s + 1.15·9-s + 0.658·10-s − 0.696·11-s + 1.27·12-s − 0.668·13-s + 0.516·14-s + 0.708·15-s − 1.11·16-s + 1.17·17-s + 1.58·18-s + 1.52·19-s + 0.417·20-s + 0.555·21-s − 0.951·22-s − 1.59·23-s − 0.271·24-s − 0.767·25-s − 0.913·26-s + 0.234·27-s + 0.326·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Analytic conductor: \(12.8255\)
Root analytic conductor: \(3.58127\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(5.540698069\)
\(L(\frac12)\) \(\approx\) \(5.540698069\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 5.76e6T \)
good2 \( 1 - 494.T + 1.31e5T^{2} \)
3 \( 1 - 1.66e4T + 1.29e8T^{2} \)
5 \( 1 - 4.21e5T + 7.62e11T^{2} \)
11 \( 1 + 4.95e8T + 5.05e17T^{2} \)
13 \( 1 + 1.96e9T + 8.65e18T^{2} \)
17 \( 1 - 3.37e10T + 8.27e20T^{2} \)
19 \( 1 - 1.12e11T + 5.48e21T^{2} \)
23 \( 1 + 6.00e11T + 1.41e23T^{2} \)
29 \( 1 - 3.81e12T + 7.25e24T^{2} \)
31 \( 1 + 8.02e12T + 2.25e25T^{2} \)
37 \( 1 + 7.74e12T + 4.56e26T^{2} \)
41 \( 1 + 3.42e13T + 2.61e27T^{2} \)
43 \( 1 - 7.74e13T + 5.87e27T^{2} \)
47 \( 1 - 6.72e13T + 2.66e28T^{2} \)
53 \( 1 + 9.80e13T + 2.05e29T^{2} \)
59 \( 1 + 1.06e15T + 1.27e30T^{2} \)
61 \( 1 - 9.02e14T + 2.24e30T^{2} \)
67 \( 1 - 4.23e15T + 1.10e31T^{2} \)
71 \( 1 - 5.31e15T + 2.96e31T^{2} \)
73 \( 1 - 8.70e15T + 4.74e31T^{2} \)
79 \( 1 - 1.22e16T + 1.81e32T^{2} \)
83 \( 1 - 2.26e16T + 4.21e32T^{2} \)
89 \( 1 + 3.04e16T + 1.37e33T^{2} \)
97 \( 1 - 3.42e16T + 5.95e33T^{2} \)
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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

−18.20966438750680626135991576272, −15.69950604696885789532890732912, −14.29218440916151780168783712618, −13.83452265447715630686918709058, −12.27485748064630261389260101259, −9.670517052175618808912621630980, −7.78839878092380633981637208483, −5.37481161991547022321889131492, −3.54114443664140139617977881879, −2.25172750830098457147843450052, 2.25172750830098457147843450052, 3.54114443664140139617977881879, 5.37481161991547022321889131492, 7.78839878092380633981637208483, 9.670517052175618808912621630980, 12.27485748064630261389260101259, 13.83452265447715630686918709058, 14.29218440916151780168783712618, 15.69950604696885789532890732912, 18.20966438750680626135991576272

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