Properties

Label 2-7-1.1-c9-0-4
Degree $2$
Conductor $7$
Sign $-1$
Analytic cond. $3.60525$
Root an. cond. $1.89874$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.8·2-s − 195.·3-s − 393.·4-s + 200.·5-s − 2.13e3·6-s − 2.40e3·7-s − 9.86e3·8-s + 1.86e4·9-s + 2.18e3·10-s + 6.38e4·11-s + 7.70e4·12-s − 1.64e5·13-s − 2.61e4·14-s − 3.93e4·15-s + 9.39e4·16-s − 3.62e5·17-s + 2.03e5·18-s − 4.36e5·19-s − 7.89e4·20-s + 4.70e5·21-s + 6.95e5·22-s + 9.18e5·23-s + 1.93e6·24-s − 1.91e6·25-s − 1.79e6·26-s + 2.00e5·27-s + 9.44e5·28-s + ⋯
L(s)  = 1  + 0.481·2-s − 1.39·3-s − 0.768·4-s + 0.143·5-s − 0.671·6-s − 0.377·7-s − 0.851·8-s + 0.948·9-s + 0.0691·10-s + 1.31·11-s + 1.07·12-s − 1.59·13-s − 0.181·14-s − 0.200·15-s + 0.358·16-s − 1.05·17-s + 0.456·18-s − 0.768·19-s − 0.110·20-s + 0.527·21-s + 0.633·22-s + 0.684·23-s + 1.18·24-s − 0.979·25-s − 0.769·26-s + 0.0724·27-s + 0.290·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-1$
Analytic conductor: \(3.60525\)
Root analytic conductor: \(1.89874\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 2.40e3T \)
good2 \( 1 - 10.8T + 512T^{2} \)
3 \( 1 + 195.T + 1.96e4T^{2} \)
5 \( 1 - 200.T + 1.95e6T^{2} \)
11 \( 1 - 6.38e4T + 2.35e9T^{2} \)
13 \( 1 + 1.64e5T + 1.06e10T^{2} \)
17 \( 1 + 3.62e5T + 1.18e11T^{2} \)
19 \( 1 + 4.36e5T + 3.22e11T^{2} \)
23 \( 1 - 9.18e5T + 1.80e12T^{2} \)
29 \( 1 + 3.68e6T + 1.45e13T^{2} \)
31 \( 1 - 3.47e6T + 2.64e13T^{2} \)
37 \( 1 - 1.88e7T + 1.29e14T^{2} \)
41 \( 1 - 2.40e6T + 3.27e14T^{2} \)
43 \( 1 + 1.25e7T + 5.02e14T^{2} \)
47 \( 1 + 5.54e7T + 1.11e15T^{2} \)
53 \( 1 + 9.26e7T + 3.29e15T^{2} \)
59 \( 1 + 2.52e7T + 8.66e15T^{2} \)
61 \( 1 - 6.93e7T + 1.16e16T^{2} \)
67 \( 1 + 2.33e7T + 2.72e16T^{2} \)
71 \( 1 + 1.06e8T + 4.58e16T^{2} \)
73 \( 1 + 2.10e8T + 5.88e16T^{2} \)
79 \( 1 + 1.49e5T + 1.19e17T^{2} \)
83 \( 1 - 5.21e8T + 1.86e17T^{2} \)
89 \( 1 - 2.98e8T + 3.50e17T^{2} \)
97 \( 1 + 8.95e8T + 7.60e17T^{2} \)
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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

−19.40423647552335104573695202448, −17.65173115358181152264424133471, −16.91114630353643452526918174418, −14.76812558417555503843247079453, −12.91204032733935345221037255583, −11.63042121779921486607189127837, −9.554205566700972309178900960496, −6.33630525032808966517598652196, −4.61918990308597522928980298957, 0, 4.61918990308597522928980298957, 6.33630525032808966517598652196, 9.554205566700972309178900960496, 11.63042121779921486607189127837, 12.91204032733935345221037255583, 14.76812558417555503843247079453, 16.91114630353643452526918174418, 17.65173115358181152264424133471, 19.40423647552335104573695202448

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