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

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.804 - 0.593i$
motivic weight  =  \(5\)
character  :  $\chi_{5} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :5/2),\ 0.804 - 0.593i)$
$L(3)$  $\approx$  $0.878396 + 0.288727i$
$L(\frac12)$  $\approx$  $0.878396 + 0.288727i$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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