Properties

Degree 2
Conductor 7
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.725·2-s + 19.6·3-s − 31.4·4-s − 46.7·5-s + 14.2·6-s + 49·7-s − 46.0·8-s + 143.·9-s − 33.8·10-s + 666.·11-s − 618.·12-s − 650.·13-s + 35.5·14-s − 918.·15-s + 973.·16-s + 1.18e3·17-s + 103.·18-s − 1.56e3·19-s + 1.47e3·20-s + 962.·21-s + 482.·22-s − 1.10e3·23-s − 904.·24-s − 939.·25-s − 471.·26-s − 1.96e3·27-s − 1.54e3·28-s + ⋯
L(s)  = 1  + 0.128·2-s + 1.26·3-s − 0.983·4-s − 0.836·5-s + 0.161·6-s + 0.377·7-s − 0.254·8-s + 0.588·9-s − 0.107·10-s + 1.65·11-s − 1.23·12-s − 1.06·13-s + 0.0484·14-s − 1.05·15-s + 0.950·16-s + 0.996·17-s + 0.0754·18-s − 0.994·19-s + 0.822·20-s + 0.476·21-s + 0.212·22-s − 0.433·23-s − 0.320·24-s − 0.300·25-s − 0.136·26-s − 0.518·27-s − 0.371·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{7} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.21406\)
\(L(\frac12)\)  \(\approx\)  \(1.21406\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 7$,\(F_p(T)\) is a polynomial of degree 2. If $p = 7$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - 49T \)
good2 \( 1 - 0.725T + 32T^{2} \)
3 \( 1 - 19.6T + 243T^{2} \)
5 \( 1 + 46.7T + 3.12e3T^{2} \)
11 \( 1 - 666.T + 1.61e5T^{2} \)
13 \( 1 + 650.T + 3.71e5T^{2} \)
17 \( 1 - 1.18e3T + 1.41e6T^{2} \)
19 \( 1 + 1.56e3T + 2.47e6T^{2} \)
23 \( 1 + 1.10e3T + 6.43e6T^{2} \)
29 \( 1 - 2.39e3T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 1.07e3T + 6.93e7T^{2} \)
41 \( 1 - 1.09e3T + 1.15e8T^{2} \)
43 \( 1 - 1.65e4T + 1.47e8T^{2} \)
47 \( 1 + 8.29e3T + 2.29e8T^{2} \)
53 \( 1 - 5.51e3T + 4.18e8T^{2} \)
59 \( 1 + 1.42e4T + 7.14e8T^{2} \)
61 \( 1 + 1.42e4T + 8.44e8T^{2} \)
67 \( 1 - 1.97e4T + 1.35e9T^{2} \)
71 \( 1 - 6.45e4T + 1.80e9T^{2} \)
73 \( 1 - 2.85e4T + 2.07e9T^{2} \)
79 \( 1 + 3.06e4T + 3.07e9T^{2} \)
83 \( 1 + 675.T + 3.93e9T^{2} \)
89 \( 1 - 1.25e5T + 5.58e9T^{2} \)
97 \( 1 + 2.29e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.56672838838674294162066102691, −19.81929608229678828534900597185, −19.11442612346681214480330983210, −17.17054155342088494212884131690, −14.82977564781548877745068927308, −14.16894755224873196216556425057, −12.20767268553459151312140921785, −9.368070419283983648535942813665, −8.013647175360602928869050064315, −3.97604672082062983728337585887, 3.97604672082062983728337585887, 8.013647175360602928869050064315, 9.368070419283983648535942813665, 12.20767268553459151312140921785, 14.16894755224873196216556425057, 14.82977564781548877745068927308, 17.17054155342088494212884131690, 19.11442612346681214480330983210, 19.81929608229678828534900597185, 21.56672838838674294162066102691

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