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Properties

Label 2-20-20.19-c6-0-5
Degree $2$
Conductor $20$
Sign $0.986 + 0.161i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.94 − 5.35i)2-s + 40.3·3-s + (6.65 + 63.6i)4-s + (−32.9 + 120. i)5-s + (−239. − 216. i)6-s + 450.·7-s + (301. − 413. i)8-s + 900.·9-s + (841. − 540. i)10-s + 390. i·11-s + (268. + 2.56e3i)12-s − 3.23e3i·13-s + (−2.67e3 − 2.41e3i)14-s + (−1.32e3 + 4.86e3i)15-s + (−4.00e3 + 846. i)16-s + 4.93e3i·17-s + ⋯
L(s)  = 1  + (−0.742 − 0.669i)2-s + 1.49·3-s + (0.103 + 0.994i)4-s + (−0.263 + 0.964i)5-s + (−1.11 − 1.00i)6-s + 1.31·7-s + (0.588 − 0.808i)8-s + 1.23·9-s + (0.841 − 0.540i)10-s + 0.293i·11-s + (0.155 + 1.48i)12-s − 1.47i·13-s + (−0.976 − 0.879i)14-s + (−0.393 + 1.44i)15-s + (−0.978 + 0.206i)16-s + 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.986 + 0.161i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ 0.986 + 0.161i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.71487 - 0.139588i\)
\(L(\frac12)\) \(\approx\) \(1.71487 - 0.139588i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.94 + 5.35i)T \)
5 \( 1 + (32.9 - 120. i)T \)
good3 \( 1 - 40.3T + 729T^{2} \)
7 \( 1 - 450.T + 1.17e5T^{2} \)
11 \( 1 - 390. iT - 1.77e6T^{2} \)
13 \( 1 + 3.23e3iT - 4.82e6T^{2} \)
17 \( 1 - 4.93e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.98e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.07e3T + 1.48e8T^{2} \)
29 \( 1 + 2.11e4T + 5.94e8T^{2} \)
31 \( 1 + 3.32e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.41e4iT - 2.56e9T^{2} \)
41 \( 1 + 4.28e4T + 4.75e9T^{2} \)
43 \( 1 - 5.77e4T + 6.32e9T^{2} \)
47 \( 1 + 1.38e5T + 1.07e10T^{2} \)
53 \( 1 + 2.77e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.81e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.40e4T + 5.15e10T^{2} \)
67 \( 1 - 2.30e5T + 9.04e10T^{2} \)
71 \( 1 - 2.14e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.11e4iT - 1.51e11T^{2} \)
79 \( 1 - 3.65e5iT - 2.43e11T^{2} \)
83 \( 1 - 5.02e4T + 3.26e11T^{2} \)
89 \( 1 + 1.77e5T + 4.96e11T^{2} \)
97 \( 1 - 1.15e6iT - 8.32e11T^{2} \)
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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

−17.48558889661022772947741409457, −15.28571890088882833489605909923, −14.54431817269777980509170120514, −13.02341124788624426655320391799, −11.23043391219707172870624015551, −10.02364768693468151543181202453, −8.295639717483761550693165813910, −7.66995808899853303094859243966, −3.61179680761534528230518901371, −2.09864674876286828660954322983, 1.66357158372556717448451033803, 4.73027334702534015785004230574, 7.47666494652404649424886342931, 8.585630213566440422903619952835, 9.297865572037654482682351096421, 11.47088192865495625452192777726, 13.73908591421366802022684977371, 14.48354468557365407100255874168, 15.72604477924345506259237773708, 16.91848936653860324603524038989

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