Properties

Label 2-7-1.1-c9-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 41.8·2-s + 0.232·3-s + 1.23e3·4-s − 1.79e3·5-s + 9.71·6-s + 2.40e3·7-s + 3.02e4·8-s − 1.96e4·9-s − 7.49e4·10-s + 1.74e4·11-s + 287.·12-s − 1.22e5·13-s + 1.00e5·14-s − 416.·15-s + 6.31e5·16-s + 3.31e5·17-s − 8.22e5·18-s + 7.61e5·19-s − 2.21e6·20-s + 557.·21-s + 7.27e5·22-s + 1.23e6·23-s + 7.02e3·24-s + 1.25e6·25-s − 5.12e6·26-s − 9.14e3·27-s + 2.96e6·28-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.00165·3-s + 2.41·4-s − 1.28·5-s + 0.00305·6-s + 0.377·7-s + 2.61·8-s − 0.999·9-s − 2.36·10-s + 0.358·11-s + 0.00399·12-s − 1.18·13-s + 0.698·14-s − 0.00212·15-s + 2.40·16-s + 0.963·17-s − 1.84·18-s + 1.34·19-s − 3.09·20-s + 0.000625·21-s + 0.662·22-s + 0.918·23-s + 0.00432·24-s + 0.643·25-s − 2.19·26-s − 0.00331·27-s + 0.911·28-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: \(0\)
Selberg data: \((2,\ 7,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(3.170984437\)
\(L(\frac12)\) \(\approx\) \(3.170984437\)
\(L(\frac{11}{2})\) 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 \( 1 - 41.8T + 512T^{2} \)
3 \( 1 - 0.232T + 1.96e4T^{2} \)
5 \( 1 + 1.79e3T + 1.95e6T^{2} \)
11 \( 1 - 1.74e4T + 2.35e9T^{2} \)
13 \( 1 + 1.22e5T + 1.06e10T^{2} \)
17 \( 1 - 3.31e5T + 1.18e11T^{2} \)
19 \( 1 - 7.61e5T + 3.22e11T^{2} \)
23 \( 1 - 1.23e6T + 1.80e12T^{2} \)
29 \( 1 - 6.34e5T + 1.45e13T^{2} \)
31 \( 1 + 5.38e6T + 2.64e13T^{2} \)
37 \( 1 + 3.03e6T + 1.29e14T^{2} \)
41 \( 1 + 7.37e6T + 3.27e14T^{2} \)
43 \( 1 + 2.06e7T + 5.02e14T^{2} \)
47 \( 1 - 2.03e7T + 1.11e15T^{2} \)
53 \( 1 + 5.97e7T + 3.29e15T^{2} \)
59 \( 1 - 6.03e7T + 8.66e15T^{2} \)
61 \( 1 + 9.44e6T + 1.16e16T^{2} \)
67 \( 1 + 2.19e8T + 2.72e16T^{2} \)
71 \( 1 + 5.58e7T + 4.58e16T^{2} \)
73 \( 1 - 4.54e8T + 5.88e16T^{2} \)
79 \( 1 - 4.51e7T + 1.19e17T^{2} \)
83 \( 1 + 3.34e8T + 1.86e17T^{2} \)
89 \( 1 - 6.51e8T + 3.50e17T^{2} \)
97 \( 1 + 1.42e9T + 7.60e17T^{2} \)
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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

−20.50428634652050050324328134539, −19.56118277473468770592031451318, −16.62776829643021273195390578558, −15.13086542264021621558258926276, −14.20223157513765993123281144803, −12.22064752638243949455290575120, −11.41703236658857597520016080649, −7.45667248779142603474364026934, −5.13978899189559338188682326891, −3.30068204782290139935686375941, 3.30068204782290139935686375941, 5.13978899189559338188682326891, 7.45667248779142603474364026934, 11.41703236658857597520016080649, 12.22064752638243949455290575120, 14.20223157513765993123281144803, 15.13086542264021621558258926276, 16.62776829643021273195390578558, 19.56118277473468770592031451318, 20.50428634652050050324328134539

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