Properties

Degree $2$
Conductor $5$
Sign $0.804 + 0.593i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.63i·2-s + 19.8i·3-s − 12·4-s + (−45 − 33.1i)5-s + 132·6-s + 59.6i·7-s − 132. i·8-s − 153·9-s + (−220. + 298. i)10-s + 252·11-s − 238. i·12-s + 119. i·13-s + 396·14-s + (660 − 895. i)15-s − 1.26e3·16-s + 689. i·17-s + ⋯
L(s)  = 1  − 1.17i·2-s + 1.27i·3-s − 0.375·4-s + (−0.804 − 0.593i)5-s + 1.49·6-s + 0.460i·7-s − 0.732i·8-s − 0.629·9-s + (−0.695 + 0.943i)10-s + 0.627·11-s − 0.478i·12-s + 0.195i·13-s + 0.539·14-s + (0.757 − 1.02i)15-s − 1.23·16-s + 0.578i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.804 + 0.593i$
Motivic weight: \(5\)
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :5/2),\ 0.804 + 0.593i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.878396 - 0.288727i\)
\(L(\frac12)\) \(\approx\) \(0.878396 - 0.288727i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (45 + 33.1i)T \)
good2 \( 1 + 6.63iT - 32T^{2} \)
3 \( 1 - 19.8iT - 243T^{2} \)
7 \( 1 - 59.6iT - 1.68e4T^{2} \)
11 \( 1 - 252T + 1.61e5T^{2} \)
13 \( 1 - 119. iT - 3.71e5T^{2} \)
17 \( 1 - 689. iT - 1.41e6T^{2} \)
19 \( 1 + 220T + 2.47e6T^{2} \)
23 \( 1 + 2.43e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.93e3T + 2.05e7T^{2} \)
31 \( 1 - 6.75e3T + 2.86e7T^{2} \)
37 \( 1 + 1.39e4iT - 6.93e7T^{2} \)
41 \( 1 + 198T + 1.15e8T^{2} \)
43 \( 1 - 417. iT - 1.47e8T^{2} \)
47 \( 1 - 1.05e4iT - 2.29e8T^{2} \)
53 \( 1 - 5.82e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.46e4T + 7.14e8T^{2} \)
61 \( 1 + 5.69e3T + 8.44e8T^{2} \)
67 \( 1 - 4.36e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.33e4T + 1.80e9T^{2} \)
73 \( 1 + 7.09e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.19e4T + 3.07e9T^{2} \)
83 \( 1 - 6.18e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.99e3T + 5.58e9T^{2} \)
97 \( 1 - 1.01e5iT - 8.58e9T^{2} \)
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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

−22.52166057003992423437663482243, −21.35581585461900188282141236080, −20.36999132347011813021972513358, −19.11732653065298533820811683384, −16.48826307049993962111149144423, −15.22209378099097969216612692439, −12.38331049598517745598077461804, −10.93869529992611551262793030092, −9.223229959911089700441868929878, −4.06350745703308212762258906706, 6.71631094869763221321108390169, 7.80950001073798191226471943289, 11.66134790726682541804104153830, 13.79103441325280814819768052363, 15.33533538841285802399005921429, 17.09272587952248977110949467303, 18.56236613057734487804330711583, 19.96105957672780268456689498558, 22.80613211102536782471174831139, 23.78022987484324488734829947865

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