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Properties

Label 2-5-5.3-c20-0-8
Degree $2$
Conductor $5$
Sign $-0.983 - 0.178i$
Analytic cond. $12.6756$
Root an. cond. $3.56029$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40e3 − 1.40e3i)2-s + (−2.31e3 − 2.31e3i)3-s − 2.87e6i·4-s + (−9.08e6 + 3.57e6i)5-s − 6.47e6·6-s + (−1.18e8 + 1.18e8i)7-s + (−2.55e9 − 2.55e9i)8-s − 3.47e9i·9-s + (−7.72e9 + 1.77e10i)10-s + 2.37e10·11-s + (−6.63e9 + 6.63e9i)12-s + (−3.84e10 − 3.84e10i)13-s + 3.32e11i·14-s + (2.92e10 + 1.27e10i)15-s − 4.14e12·16-s + (1.43e12 − 1.43e12i)17-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (−0.0391 − 0.0391i)3-s − 2.74i·4-s + (−0.930 + 0.365i)5-s − 0.107·6-s + (−0.420 + 0.420i)7-s + (−2.37 − 2.37i)8-s − 0.996i·9-s + (−0.772 + 1.77i)10-s + 0.917·11-s + (−0.107 + 0.107i)12-s + (−0.279 − 0.279i)13-s + 1.14i·14-s + (0.0507 + 0.0221i)15-s − 3.76·16-s + (0.710 − 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.983 - 0.178i$
Analytic conductor: \(12.6756\)
Root analytic conductor: \(3.56029\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :10),\ -0.983 - 0.178i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.225500 + 2.50826i\)
\(L(\frac12)\) \(\approx\) \(0.225500 + 2.50826i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (9.08e6 - 3.57e6i)T \)
good2 \( 1 + (-1.40e3 + 1.40e3i)T - 1.04e6iT^{2} \)
3 \( 1 + (2.31e3 + 2.31e3i)T + 3.48e9iT^{2} \)
7 \( 1 + (1.18e8 - 1.18e8i)T - 7.97e16iT^{2} \)
11 \( 1 - 2.37e10T + 6.72e20T^{2} \)
13 \( 1 + (3.84e10 + 3.84e10i)T + 1.90e22iT^{2} \)
17 \( 1 + (-1.43e12 + 1.43e12i)T - 4.06e24iT^{2} \)
19 \( 1 - 3.57e12iT - 3.75e25T^{2} \)
23 \( 1 + (1.53e13 + 1.53e13i)T + 1.71e27iT^{2} \)
29 \( 1 + 6.21e14iT - 1.76e29T^{2} \)
31 \( 1 - 1.14e15T + 6.71e29T^{2} \)
37 \( 1 + (-2.42e15 + 2.42e15i)T - 2.31e31iT^{2} \)
41 \( 1 + 2.09e15T + 1.80e32T^{2} \)
43 \( 1 + (6.21e15 + 6.21e15i)T + 4.67e32iT^{2} \)
47 \( 1 + (5.36e16 - 5.36e16i)T - 2.76e33iT^{2} \)
53 \( 1 + (-1.43e17 - 1.43e17i)T + 3.05e34iT^{2} \)
59 \( 1 - 1.76e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.15e17T + 5.08e35T^{2} \)
67 \( 1 + (-6.17e17 + 6.17e17i)T - 3.32e36iT^{2} \)
71 \( 1 - 1.37e18T + 1.05e37T^{2} \)
73 \( 1 + (-1.21e18 - 1.21e18i)T + 1.84e37iT^{2} \)
79 \( 1 - 7.66e18iT - 8.96e37T^{2} \)
83 \( 1 + (-4.96e18 - 4.96e18i)T + 2.40e38iT^{2} \)
89 \( 1 + 3.67e19iT - 9.72e38T^{2} \)
97 \( 1 + (-4.64e19 + 4.64e19i)T - 5.43e39iT^{2} \)
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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

−18.67136579924787017785257352418, −15.42481408275737312669486975201, −14.30357350727850247946850209466, −12.34716394707083497103341901651, −11.66252950527205227814519254221, −9.780385144218787488175530830977, −6.22440002826388464022208462630, −4.12556489554106639324742659653, −2.92328865670922642446819002790, −0.72553998038403779295825759939, 3.60142061856742226251512904790, 4.90331831494468946265393987004, 6.84251777133775110881570182560, 8.212781375383566670526321783816, 11.86367932372194830763485500070, 13.28666893221336689884783428155, 14.71515030253356794014867142925, 16.15849993988063013879657054400, 16.94430615562688042139628237860, 19.73282001256587183585042047080

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