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Properties

Label 2-7-7.6-c12-0-1
Degree $2$
Conductor $7$
Sign $-0.837 + 0.546i$
Analytic cond. $6.39795$
Root an. cond. $2.52941$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 108.·2-s + 1.04e3i·3-s + 7.71e3·4-s + 2.32e4i·5-s − 1.13e5i·6-s + (−9.85e4 + 6.42e4i)7-s − 3.92e5·8-s − 5.67e5·9-s − 2.52e6i·10-s + 1.17e6·11-s + 8.08e6i·12-s − 1.30e6i·13-s + (1.07e7 − 6.98e6i)14-s − 2.43e7·15-s + 1.10e7·16-s + 1.42e7i·17-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.43i·3-s + 1.88·4-s + 1.48i·5-s − 2.44i·6-s + (−0.837 + 0.546i)7-s − 1.49·8-s − 1.06·9-s − 2.52i·10-s + 0.660·11-s + 2.70i·12-s − 0.269i·13-s + (1.42 − 0.927i)14-s − 2.14·15-s + 0.661·16-s + 0.592i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-0.837 + 0.546i$
Analytic conductor: \(6.39795\)
Root analytic conductor: \(2.52941\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :6),\ -0.837 + 0.546i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.135126 - 0.454557i\)
\(L(\frac12)\) \(\approx\) \(0.135126 - 0.454557i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (9.85e4 - 6.42e4i)T \)
good2 \( 1 + 108.T + 4.09e3T^{2} \)
3 \( 1 - 1.04e3iT - 5.31e5T^{2} \)
5 \( 1 - 2.32e4iT - 2.44e8T^{2} \)
11 \( 1 - 1.17e6T + 3.13e12T^{2} \)
13 \( 1 + 1.30e6iT - 2.32e13T^{2} \)
17 \( 1 - 1.42e7iT - 5.82e14T^{2} \)
19 \( 1 + 5.15e7iT - 2.21e15T^{2} \)
23 \( 1 - 1.32e8T + 2.19e16T^{2} \)
29 \( 1 - 1.43e8T + 3.53e17T^{2} \)
31 \( 1 - 6.22e8iT - 7.87e17T^{2} \)
37 \( 1 - 5.77e8T + 6.58e18T^{2} \)
41 \( 1 + 3.87e9iT - 2.25e19T^{2} \)
43 \( 1 + 1.05e10T + 3.99e19T^{2} \)
47 \( 1 - 1.69e10iT - 1.16e20T^{2} \)
53 \( 1 + 1.48e10T + 4.91e20T^{2} \)
59 \( 1 - 6.12e10iT - 1.77e21T^{2} \)
61 \( 1 + 4.42e10iT - 2.65e21T^{2} \)
67 \( 1 - 1.23e11T + 8.18e21T^{2} \)
71 \( 1 + 1.23e11T + 1.64e22T^{2} \)
73 \( 1 - 1.13e11iT - 2.29e22T^{2} \)
79 \( 1 - 1.22e11T + 5.90e22T^{2} \)
83 \( 1 + 1.49e11iT - 1.06e23T^{2} \)
89 \( 1 + 2.82e11iT - 2.46e23T^{2} \)
97 \( 1 + 7.83e11iT - 6.93e23T^{2} \)
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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

−19.80040756866974242378006249658, −18.76720744764010379114817154838, −17.32583570772241401635961813066, −15.88233693852149658308350169369, −14.97198602072170852525448934111, −11.13432419732506562930001529874, −10.16273159176627102022328137819, −9.058900317449407020494836251191, −6.71482243616407418355749857588, −3.02252544791014951507933902117, 0.48719830944483636848536498886, 1.48632692261639755718616783410, 6.73721542705346896210666144762, 8.177421752914641306864758918469, 9.538049122671531084053366233196, 11.93271447331547792314939284833, 13.25985048575423552419965476171, 16.42482757478944971968175251198, 17.10871782792205431785628801682, 18.57573000008270808749098421053

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