Properties

Degree $2$
Conductor $5$
Sign $-0.969 + 0.246i$
Motivic weight $8$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.8 + 11.8i)2-s + (−90.9 − 90.9i)3-s − 25.5i·4-s + (−434. + 449. i)5-s + 2.15e3·6-s + (508. − 508. i)7-s + (−2.73e3 − 2.73e3i)8-s + 9.98e3i·9-s + (−178. − 1.04e4i)10-s − 7.02e3·11-s + (−2.32e3 + 2.32e3i)12-s + (−8.07e3 − 8.07e3i)13-s + 1.20e4i·14-s + (8.03e4 − 1.36e3i)15-s + 7.14e4·16-s + (−1.02e5 + 1.02e5i)17-s + ⋯
L(s)  = 1  + (−0.741 + 0.741i)2-s + (−1.12 − 1.12i)3-s − 0.0998i·4-s + (−0.694 + 0.719i)5-s + 1.66·6-s + (0.211 − 0.211i)7-s + (−0.667 − 0.667i)8-s + 1.52i·9-s + (−0.0178 − 1.04i)10-s − 0.479·11-s + (−0.112 + 0.112i)12-s + (−0.282 − 0.282i)13-s + 0.313i·14-s + (1.58 − 0.0269i)15-s + 1.08·16-s + (−1.23 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.969 + 0.246i$
Motivic weight: \(8\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :4),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.00665038 - 0.0531824i\)
\(L(\frac12)\) \(\approx\) \(0.00665038 - 0.0531824i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (434. - 449. i)T \)
good2 \( 1 + (11.8 - 11.8i)T - 256iT^{2} \)
3 \( 1 + (90.9 + 90.9i)T + 6.56e3iT^{2} \)
7 \( 1 + (-508. + 508. i)T - 5.76e6iT^{2} \)
11 \( 1 + 7.02e3T + 2.14e8T^{2} \)
13 \( 1 + (8.07e3 + 8.07e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (1.02e5 - 1.02e5i)T - 6.97e9iT^{2} \)
19 \( 1 + 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 + (-1.32e5 - 1.32e5i)T + 7.83e10iT^{2} \)
29 \( 1 + 3.92e5iT - 5.00e11T^{2} \)
31 \( 1 + 5.07e5T + 8.52e11T^{2} \)
37 \( 1 + (6.10e4 - 6.10e4i)T - 3.51e12iT^{2} \)
41 \( 1 + 1.81e6T + 7.98e12T^{2} \)
43 \( 1 + (-1.47e6 - 1.47e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (1.79e6 - 1.79e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (5.66e6 + 5.66e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 1.74e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.96e7T + 1.91e14T^{2} \)
67 \( 1 + (1.12e7 - 1.12e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 3.01e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.52e7 - 2.52e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 8.14e6iT - 1.51e15T^{2} \)
83 \( 1 + (1.99e7 + 1.99e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 8.20e7iT - 3.93e15T^{2} \)
97 \( 1 + (-3.37e7 + 3.37e7i)T - 7.83e15iT^{2} \)
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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

−23.53098567406534841193756384845, −22.08894539534873396978978554019, −19.25727307770337117688973101828, −18.04864154026151629261208143277, −17.19225213140584550836122099376, −15.46940256754163654224379282199, −12.76554466803163537011519474090, −11.09899152288409629868316510951, −7.82131647649432496149521622480, −6.56620737840845737433603586750, 0.06809824173264623395016738569, 4.99409213513494618218204955326, 9.148120791026819019623247058122, 10.80904127711720388284476460596, 11.91800541403415463872960774073, 15.40197370464915134439909203977, 16.72310511439631781797925453966, 18.25187342918815243568489402509, 20.10087943313760051360575393936, 21.20228482606941885342018516746

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