Properties

Label 2-7-7.5-c8-0-0
Degree $2$
Conductor $7$
Sign $-0.885 + 0.464i$
Analytic cond. $2.85165$
Root an. cond. $1.68868$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−12.6 + 21.9i)2-s + (−7.68 + 4.43i)3-s + (−193. − 334. i)4-s + (−538. − 310. i)5-s − 224. i·6-s + (207. + 2.39e3i)7-s + 3.29e3·8-s + (−3.24e3 + 5.61e3i)9-s + (1.36e4 − 7.87e3i)10-s + (−7.54e3 − 1.30e4i)11-s + (2.96e3 + 1.71e3i)12-s + 4.51e4i·13-s + (−5.51e4 − 2.57e4i)14-s + 5.51e3·15-s + (7.65e3 − 1.32e4i)16-s + (3.59e4 − 2.07e4i)17-s + ⋯
L(s)  = 1  + (−0.791 + 1.37i)2-s + (−0.0948 + 0.0547i)3-s + (−0.754 − 1.30i)4-s + (−0.861 − 0.497i)5-s − 0.173i·6-s + (0.0864 + 0.996i)7-s + 0.804·8-s + (−0.494 + 0.855i)9-s + (1.36 − 0.787i)10-s + (−0.515 − 0.892i)11-s + (0.143 + 0.0825i)12-s + 1.57i·13-s + (−1.43 − 0.670i)14-s + 0.108·15-s + (0.116 − 0.202i)16-s + (0.430 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(2.85165\)
Root analytic conductor: \(1.68868\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :4),\ -0.885 + 0.464i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0996243 - 0.404143i\)
\(L(\frac12)\) \(\approx\) \(0.0996243 - 0.404143i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-207. - 2.39e3i)T \)
good2 \( 1 + (12.6 - 21.9i)T + (-128 - 221. i)T^{2} \)
3 \( 1 + (7.68 - 4.43i)T + (3.28e3 - 5.68e3i)T^{2} \)
5 \( 1 + (538. + 310. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (7.54e3 + 1.30e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 4.51e4iT - 8.15e8T^{2} \)
17 \( 1 + (-3.59e4 + 2.07e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.02e5 - 5.90e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.08e5 - 1.87e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 3.02e4T + 5.00e11T^{2} \)
31 \( 1 + (1.11e6 - 6.42e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.31e6 + 2.27e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 1.05e6iT - 7.98e12T^{2} \)
43 \( 1 + 6.68e5T + 1.16e13T^{2} \)
47 \( 1 + (-8.46e5 - 4.88e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-2.36e5 - 4.08e5i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (1.62e7 - 9.35e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-2.08e7 - 1.20e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-5.58e6 - 9.67e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 7.52e6T + 6.45e14T^{2} \)
73 \( 1 + (-8.54e6 + 4.93e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (7.33e6 - 1.27e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 3.82e6iT - 2.25e15T^{2} \)
89 \( 1 + (-3.43e7 - 1.98e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 7.73e7iT - 7.83e15T^{2} \)
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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

−21.59762731525319911731407686899, −19.37859080955687407921556806281, −18.35586173558354457926947050254, −16.43503006301631741170493574567, −16.01997352000235736976215571779, −14.26079677183159962009615069588, −11.68657371086322816036152994880, −9.007986940173629443759745171389, −7.81044747441236046957725910062, −5.53847424697892715859035240096, 0.42677902186565970797190865179, 3.36332384515845514557858091343, 7.78651959557278014319826916202, 10.01502500553811457196240400350, 11.25367809818783152389041622299, 12.67606891811590983153740405985, 15.10162438366285117828757996645, 17.41728519699956085515454724301, 18.43815482799568929065553527631, 20.13293929374421321854614412568

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