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Properties

Label 2-5-5.3-c10-0-1
Degree $2$
Conductor $5$
Sign $-0.315 + 0.949i$
Analytic cond. $3.17678$
Root an. cond. $1.78235$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−36.6 + 36.6i)2-s + (−13.8 − 13.8i)3-s − 1.66e3i·4-s + (−721. − 3.04e3i)5-s + 1.01e3·6-s + (−1.38e4 + 1.38e4i)7-s + (2.33e4 + 2.33e4i)8-s − 5.86e4i·9-s + (1.37e5 + 8.49e4i)10-s − 2.57e5·11-s + (−2.29e4 + 2.29e4i)12-s + (1.07e5 + 1.07e5i)13-s − 1.01e6i·14-s + (−3.20e4 + 5.20e4i)15-s − 1.12e4·16-s + (−4.39e5 + 4.39e5i)17-s + ⋯
L(s)  = 1  + (−1.14 + 1.14i)2-s + (−0.0569 − 0.0569i)3-s − 1.62i·4-s + (−0.230 − 0.973i)5-s + 0.130·6-s + (−0.826 + 0.826i)7-s + (0.713 + 0.713i)8-s − 0.993i·9-s + (1.37 + 0.849i)10-s − 1.59·11-s + (−0.0923 + 0.0923i)12-s + (0.288 + 0.288i)13-s − 1.89i·14-s + (−0.0422 + 0.0685i)15-s − 0.0107·16-s + (−0.309 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.315 + 0.949i$
Analytic conductor: \(3.17678\)
Root analytic conductor: \(1.78235\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :5),\ -0.315 + 0.949i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0725546 - 0.100553i\)
\(L(\frac12)\) \(\approx\) \(0.0725546 - 0.100553i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (721. + 3.04e3i)T \)
good2 \( 1 + (36.6 - 36.6i)T - 1.02e3iT^{2} \)
3 \( 1 + (13.8 + 13.8i)T + 5.90e4iT^{2} \)
7 \( 1 + (1.38e4 - 1.38e4i)T - 2.82e8iT^{2} \)
11 \( 1 + 2.57e5T + 2.59e10T^{2} \)
13 \( 1 + (-1.07e5 - 1.07e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (4.39e5 - 4.39e5i)T - 2.01e12iT^{2} \)
19 \( 1 - 1.46e6iT - 6.13e12T^{2} \)
23 \( 1 + (3.17e6 + 3.17e6i)T + 4.14e13iT^{2} \)
29 \( 1 + 1.87e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.47e7T + 8.19e14T^{2} \)
37 \( 1 + (-1.77e7 + 1.77e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.46e8T + 1.34e16T^{2} \)
43 \( 1 + (8.02e7 + 8.02e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.20e8 - 1.20e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (4.48e8 + 4.48e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 2.00e8iT - 5.11e17T^{2} \)
61 \( 1 + 4.13e7T + 7.13e17T^{2} \)
67 \( 1 + (-1.80e8 + 1.80e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 3.07e8T + 3.25e18T^{2} \)
73 \( 1 + (1.88e9 + 1.88e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 3.83e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.47e9 - 2.47e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 3.22e9iT - 3.11e19T^{2} \)
97 \( 1 + (2.24e8 - 2.24e8i)T - 7.37e19iT^{2} \)
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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

−20.91786949440419291049022172277, −19.03972873461781291980663339671, −17.85419465038530680595408698562, −16.22830535335301902730046224698, −15.42450014766179170428202850477, −12.61882099514438021896139125609, −9.605202129515795081405051467158, −8.238369916352422358927184627645, −6.00538086694833679324473316969, −0.12884671991971057629268711482, 2.82766553468071377367811853355, 7.69740493281476003312883576578, 10.15793776436543915289126541476, 10.98888133107929397842395333287, 13.28877606597857249822928924492, 16.05313019702372243812113042010, 17.90763000311153116120025609369, 19.06066998685258030529865929454, 20.12081285586125728249426684752, 21.76839894179693404927698170272

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