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Properties

Label 2-90-5.2-c6-0-8
Degree $2$
Conductor $90$
Sign $0.986 + 0.162i$
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 + (−124. + 8.53i)5-s + (−288. − 288. i)7-s + (−128 + 128i)8-s + (−532. − 464. i)10-s + 2.63e3·11-s + (1.48e3 − 1.48e3i)13-s − 2.31e3i·14-s − 1.02e3·16-s + (4.91e3 + 4.91e3i)17-s − 6.98e3i·19-s + (−273. − 3.99e3i)20-s + (1.05e4 + 1.05e4i)22-s + (−3.66e3 + 3.66e3i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.997 + 0.0682i)5-s + (−0.842 − 0.842i)7-s + (−0.250 + 0.250i)8-s + (−0.532 − 0.464i)10-s + 1.98·11-s + (0.675 − 0.675i)13-s − 0.842i·14-s − 0.250·16-s + (1.00 + 1.00i)17-s − 1.01i·19-s + (−0.0341 − 0.498i)20-s + (0.990 + 0.990i)22-s + (−0.300 + 0.300i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.986 + 0.162i$
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: \((2,\ 90,\ (\ :3),\ 0.986 + 0.162i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.98009 - 0.162230i\)
\(L(\frac12)\) \(\approx\) \(1.98009 - 0.162230i\)
\(L(4)\) 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 \( 1 + (124. - 8.53i)T \)
good7 \( 1 + (288. + 288. i)T + 1.17e5iT^{2} \)
11 \( 1 - 2.63e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.48e3 + 1.48e3i)T - 4.82e6iT^{2} \)
17 \( 1 + (-4.91e3 - 4.91e3i)T + 2.41e7iT^{2} \)
19 \( 1 + 6.98e3iT - 4.70e7T^{2} \)
23 \( 1 + (3.66e3 - 3.66e3i)T - 1.48e8iT^{2} \)
29 \( 1 + 3.91e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.57e4T + 8.87e8T^{2} \)
37 \( 1 + (521. + 521. i)T + 2.56e9iT^{2} \)
41 \( 1 - 7.28e4T + 4.75e9T^{2} \)
43 \( 1 + (-4.75e4 + 4.75e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-438. - 438. i)T + 1.07e10iT^{2} \)
53 \( 1 + (1.86e4 - 1.86e4i)T - 2.21e10iT^{2} \)
59 \( 1 + 3.12e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.27e5T + 5.15e10T^{2} \)
67 \( 1 + (-6.93e3 - 6.93e3i)T + 9.04e10iT^{2} \)
71 \( 1 + 5.03e4T + 1.28e11T^{2} \)
73 \( 1 + (-1.67e5 + 1.67e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 2.00e5iT - 2.43e11T^{2} \)
83 \( 1 + (2.22e5 - 2.22e5i)T - 3.26e11iT^{2} \)
89 \( 1 - 7.34e5iT - 4.96e11T^{2} \)
97 \( 1 + (5.97e5 + 5.97e5i)T + 8.32e11iT^{2} \)
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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

−12.94115181001489239737091796385, −11.98312581066833105617489255749, −10.94200541648251465859375185472, −9.473404133296396735395205692905, −8.148609790865441145114407785320, −7.03122115266869074318671362887, −6.08247778373819066278293174822, −4.08167431764913110604190464394, −3.53662772485637252338322153985, −0.74478658166312553593766945911, 1.21800176615760542349478999145, 3.23037167080652057833105036683, 4.14339606733077983820595667142, 5.87607167948505986206808975648, 7.00995631733830935187862469053, 8.774855411939146991170496542242, 9.568106259968066789131015462113, 11.18064452418169020396877703259, 12.05889965423799788656701628658, 12.50451332095020660266561454844

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