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  + 8.27·2-s − 25.6·3-s + 36.4·4-s + 28.7·5-s − 212.·6-s + 49·7-s + 37.0·8-s + 414.·9-s + 237.·10-s − 270.·11-s − 935.·12-s + 300.·13-s + 405.·14-s − 737.·15-s − 860.·16-s + 613.·17-s + 3.43e3·18-s − 1.70e3·19-s + 1.04e3·20-s − 1.25e3·21-s − 2.23e3·22-s + 3.18e3·23-s − 949.·24-s − 2.29e3·25-s + 2.48e3·26-s − 4.40e3·27-s + 1.78e3·28-s + ⋯
L(s)  = 1  + 1.46·2-s − 1.64·3-s + 1.13·4-s + 0.514·5-s − 2.40·6-s + 0.377·7-s + 0.204·8-s + 1.70·9-s + 0.752·10-s − 0.673·11-s − 1.87·12-s + 0.493·13-s + 0.552·14-s − 0.846·15-s − 0.840·16-s + 0.514·17-s + 2.49·18-s − 1.08·19-s + 0.586·20-s − 0.621·21-s − 0.984·22-s + 1.25·23-s − 0.336·24-s − 0.735·25-s + 0.721·26-s − 1.16·27-s + 0.430·28-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  =  \((2,\ 7,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.41111\)
\(L(\frac12)\)  \(\approx\)  \(1.41111\)
\(L(\frac{7}{2})\)   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 \( 1 - 8.27T + 32T^{2} \)
3 \( 1 + 25.6T + 243T^{2} \)
5 \( 1 - 28.7T + 3.12e3T^{2} \)
11 \( 1 + 270.T + 1.61e5T^{2} \)
13 \( 1 - 300.T + 3.71e5T^{2} \)
17 \( 1 - 613.T + 1.41e6T^{2} \)
19 \( 1 + 1.70e3T + 2.47e6T^{2} \)
23 \( 1 - 3.18e3T + 6.43e6T^{2} \)
29 \( 1 - 4.29e3T + 2.05e7T^{2} \)
31 \( 1 - 2.02e3T + 2.86e7T^{2} \)
37 \( 1 - 5.15e3T + 6.93e7T^{2} \)
41 \( 1 + 7.14e3T + 1.15e8T^{2} \)
43 \( 1 + 1.95e4T + 1.47e8T^{2} \)
47 \( 1 - 1.99e4T + 2.29e8T^{2} \)
53 \( 1 - 3.94e3T + 4.18e8T^{2} \)
59 \( 1 + 2.97e4T + 7.14e8T^{2} \)
61 \( 1 + 5.05e4T + 8.44e8T^{2} \)
67 \( 1 - 5.05e3T + 1.35e9T^{2} \)
71 \( 1 - 3.28e4T + 1.80e9T^{2} \)
73 \( 1 + 1.11e4T + 2.07e9T^{2} \)
79 \( 1 - 8.18e4T + 3.07e9T^{2} \)
83 \( 1 - 1.18e5T + 3.93e9T^{2} \)
89 \( 1 + 4.16e4T + 5.58e9T^{2} \)
97 \( 1 - 4.36e4T + 8.58e9T^{2} \)
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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

−21.76225147842367643539943996651, −21.10005866446434830838495013222, −18.27824021515798302066668506708, −16.93896939131145108623869960687, −15.35081808140215523817312865399, −13.43779131529310651762744374699, −12.14564071685321938232185303844, −10.76195289371562611183288647496, −6.25933100890702232634318684891, −4.95482419936866029068056129325, 4.95482419936866029068056129325, 6.25933100890702232634318684891, 10.76195289371562611183288647496, 12.14564071685321938232185303844, 13.43779131529310651762744374699, 15.35081808140215523817312865399, 16.93896939131145108623869960687, 18.27824021515798302066668506708, 21.10005866446434830838495013222, 21.76225147842367643539943996651

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