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Properties

Label 2-90-5.2-c6-0-5
Degree $2$
Conductor $90$
Sign $-0.504 - 0.863i$
Analytic cond. $20.7048$
Root an. cond. $4.55026$
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  + (4 + 4i)2-s + 32i·4-s + (86.1 + 90.5i)5-s + (2.98 + 2.98i)7-s + (−128 + 128i)8-s + (−17.3 + 706. i)10-s + 2.00e3·11-s + (−2.48e3 + 2.48e3i)13-s + 23.9i·14-s − 1.02e3·16-s + (699. + 699. i)17-s − 5.28e3i·19-s + (−2.89e3 + 2.75e3i)20-s + (8.01e3 + 8.01e3i)22-s + (−8.19e3 + 8.19e3i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.689 + 0.724i)5-s + (0.00871 + 0.00871i)7-s + (−0.250 + 0.250i)8-s + (−0.0173 + 0.706i)10-s + 1.50·11-s + (−1.13 + 1.13i)13-s + 0.00871i·14-s − 0.250·16-s + (0.142 + 0.142i)17-s − 0.770i·19-s + (−0.362 + 0.344i)20-s + (0.752 + 0.752i)22-s + (−0.673 + 0.673i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(20.7048\)
Root analytic conductor: \(4.55026\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :3),\ -0.504 - 0.863i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.29632 + 2.25957i\)
\(L(\frac12)\) \(\approx\) \(1.29632 + 2.25957i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 4i)T \)
3 \( 1 \)
5 \( 1 + (-86.1 - 90.5i)T \)
good7 \( 1 + (-2.98 - 2.98i)T + 1.17e5iT^{2} \)
11 \( 1 - 2.00e3T + 1.77e6T^{2} \)
13 \( 1 + (2.48e3 - 2.48e3i)T - 4.82e6iT^{2} \)
17 \( 1 + (-699. - 699. i)T + 2.41e7iT^{2} \)
19 \( 1 + 5.28e3iT - 4.70e7T^{2} \)
23 \( 1 + (8.19e3 - 8.19e3i)T - 1.48e8iT^{2} \)
29 \( 1 - 4.73e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.21e4T + 8.87e8T^{2} \)
37 \( 1 + (1.91e4 + 1.91e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 6.67e4T + 4.75e9T^{2} \)
43 \( 1 + (4.31e4 - 4.31e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-3.87e4 - 3.87e4i)T + 1.07e10iT^{2} \)
53 \( 1 + (-1.15e5 + 1.15e5i)T - 2.21e10iT^{2} \)
59 \( 1 + 2.58e3iT - 4.21e10T^{2} \)
61 \( 1 - 2.39e5T + 5.15e10T^{2} \)
67 \( 1 + (-9.73e4 - 9.73e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 4.89e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.76e5 + 3.76e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 2.70e5iT - 2.43e11T^{2} \)
83 \( 1 + (-6.36e5 + 6.36e5i)T - 3.26e11iT^{2} \)
89 \( 1 - 2.00e5iT - 4.96e11T^{2} \)
97 \( 1 + (2.93e5 + 2.93e5i)T + 8.32e11iT^{2} \)
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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

−13.61426125715124603134447118535, −12.22483889065581675174509346470, −11.35454977500351025816301801691, −9.852828205822733479565635463812, −8.921252436835584884128650629023, −7.10441856931136775572985311723, −6.53815448105767873683383706377, −5.06139056561354945175234239273, −3.57529120227795552978774051993, −1.91948092966647814700783760470, 0.797608245228006676548514523086, 2.25657782457098327105313612017, 4.01580038619289365316511878230, 5.29880255347128199154275726398, 6.42532279837242430703810355235, 8.192430063498535768267853542526, 9.562759950814433587725911056562, 10.22534457813533570162219880787, 11.89675875263115131871177007650, 12.39606892892691954707093045749

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