Properties

Degree $2$
Conductor $5$
Sign $-0.170 - 0.985i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.54e4 + 6.54e4i)2-s + (−1.37e7 + 1.37e7i)3-s + 4.26e9i·4-s + (5.68e10 − 1.41e11i)5-s − 1.80e12·6-s + (−2.41e12 − 2.41e12i)7-s + (2.14e12 − 2.14e12i)8-s + 1.47e15i·9-s + (1.29e16 − 5.54e15i)10-s + 4.03e16·11-s + (−5.86e16 − 5.86e16i)12-s + (−2.33e17 + 2.33e17i)13-s − 3.15e17i·14-s + (1.16e18 + 2.73e18i)15-s + 1.85e19·16-s + (5.44e19 + 5.44e19i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.998i)2-s + (−0.319 + 0.319i)3-s + 0.992i·4-s + (0.372 − 0.927i)5-s − 0.638·6-s + (−0.0725 − 0.0725i)7-s + (0.00763 − 0.00763i)8-s + 0.795i·9-s + (1.29 − 0.554i)10-s + 0.878·11-s + (−0.317 − 0.317i)12-s + (−0.351 + 0.351i)13-s − 0.144i·14-s + (0.177 + 0.416i)15-s + 1.00·16-s + (1.11 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.170 - 0.985i$
Motivic weight: \(32\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.170 - 0.985i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(3.569522117\)
\(L(\frac12)\) \(\approx\) \(3.569522117\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.68e10 + 1.41e11i)T \)
good2 \( 1 + (-6.54e4 - 6.54e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (1.37e7 - 1.37e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (2.41e12 + 2.41e12i)T + 1.10e27iT^{2} \)
11 \( 1 - 4.03e16T + 2.11e33T^{2} \)
13 \( 1 + (2.33e17 - 2.33e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (-5.44e19 - 5.44e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 2.36e20iT - 8.31e40T^{2} \)
23 \( 1 + (-9.56e20 + 9.56e20i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.73e22iT - 6.26e46T^{2} \)
31 \( 1 - 2.11e23T + 5.29e47T^{2} \)
37 \( 1 + (-1.04e25 - 1.04e25i)T + 1.52e50iT^{2} \)
41 \( 1 + 3.86e25T + 4.06e51T^{2} \)
43 \( 1 + (1.15e26 - 1.15e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (-5.56e26 - 5.56e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-1.97e27 + 1.97e27i)T - 1.50e55iT^{2} \)
59 \( 1 - 2.24e28iT - 4.64e56T^{2} \)
61 \( 1 - 4.54e28T + 1.35e57T^{2} \)
67 \( 1 + (1.71e29 + 1.71e29i)T + 2.71e58iT^{2} \)
71 \( 1 - 4.36e29T + 1.73e59T^{2} \)
73 \( 1 + (-6.60e29 + 6.60e29i)T - 4.22e59iT^{2} \)
79 \( 1 + 2.22e30iT - 5.29e60T^{2} \)
83 \( 1 + (-1.54e30 + 1.54e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 2.18e31iT - 2.40e62T^{2} \)
97 \( 1 + (7.63e31 + 7.63e31i)T + 3.77e63iT^{2} \)
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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

−16.50973765619687349486760215776, −14.79676365564624465572669244453, −13.55620846546501258223630992006, −12.22897359629697252403366139590, −10.01065940532844333596790285826, −8.036611506800156996435292401089, −6.21432952027602095428320811654, −5.11415711654087642517818900936, −4.00251923188484058515618651699, −1.40161105892212653311861015923, 0.939713531706334593474959071767, 2.54657941953177809073013103486, 3.65809268242083688264966857048, 5.50924729896237456369515453237, 7.01548716269725891351861234257, 9.736683937028964440801264184674, 11.31390087059891803847542885499, 12.26273424476759464298638018330, 13.78088119459185975464988029886, 14.88615440242878141456910302643

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