Properties

Label 2-90-5.2-c4-0-2
Degree $2$
Conductor $90$
Sign $-0.400 - 0.916i$
Analytic cond. $9.30329$
Root an. cond. $3.05013$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + 8i·4-s + (15 + 20i)5-s + (−19 − 19i)7-s + (16 − 16i)8-s + (10 − 70i)10-s − 202·11-s + (−99 + 99i)13-s + 76i·14-s − 64·16-s + (239 + 239i)17-s + 40i·19-s + (−160 + 120i)20-s + (404 + 404i)22-s + (−541 + 541i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (0.599 + 0.800i)5-s + (−0.387 − 0.387i)7-s + (0.250 − 0.250i)8-s + (0.100 − 0.700i)10-s − 1.66·11-s + (−0.585 + 0.585i)13-s + 0.387i·14-s − 0.250·16-s + (0.826 + 0.826i)17-s + 0.110i·19-s + (−0.400 + 0.299i)20-s + (0.834 + 0.834i)22-s + (−1.02 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(9.30329\)
Root analytic conductor: \(3.05013\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :2),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.333954 + 0.510212i\)
\(L(\frac12)\) \(\approx\) \(0.333954 + 0.510212i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
3 \( 1 \)
5 \( 1 + (-15 - 20i)T \)
good7 \( 1 + (19 + 19i)T + 2.40e3iT^{2} \)
11 \( 1 + 202T + 1.46e4T^{2} \)
13 \( 1 + (99 - 99i)T - 2.85e4iT^{2} \)
17 \( 1 + (-239 - 239i)T + 8.35e4iT^{2} \)
19 \( 1 - 40iT - 1.30e5T^{2} \)
23 \( 1 + (541 - 541i)T - 2.79e5iT^{2} \)
29 \( 1 - 200iT - 7.07e5T^{2} \)
31 \( 1 + 758T + 9.23e5T^{2} \)
37 \( 1 + (-141 - 141i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.04e3T + 2.82e6T^{2} \)
43 \( 1 + (759 - 759i)T - 3.41e6iT^{2} \)
47 \( 1 + (-459 - 459i)T + 4.87e6iT^{2} \)
53 \( 1 + (-1.81e3 + 1.81e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 4.60e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.08e3T + 1.38e7T^{2} \)
67 \( 1 + (-5.08e3 - 5.08e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 3.47e3T + 2.54e7T^{2} \)
73 \( 1 + (3.47e3 - 3.47e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 7.68e3iT - 3.89e7T^{2} \)
83 \( 1 + (6.08e3 - 6.08e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 5.68e3iT - 6.27e7T^{2} \)
97 \( 1 + (-561 - 561i)T + 8.85e7iT^{2} \)
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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

−13.55587006766095780163999975990, −12.68361577033720823775527039842, −11.33075157845715485501597031638, −10.22557734096621359205708945152, −9.804987257250658823066328684423, −8.085342909911672044776413085157, −7.04843808649719644913597511611, −5.52396644809420505973727423818, −3.47216428170075910887704865808, −2.06784618085510976159195473279, 0.31915024077190792149422476443, 2.47843457632540443297977885334, 5.02540478444843293350695594715, 5.84004202866023806945442058512, 7.51646940517610594856224128443, 8.511558204659068927563556469107, 9.725235533062304588575160156151, 10.42437123745390542117313791808, 12.19234498042640476904480803585, 13.05231796696209738124488520596

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