Properties

Label 2-90-5.3-c10-0-23
Degree $2$
Conductor $90$
Sign $-0.858 + 0.513i$
Analytic cond. $57.1821$
Root an. cond. $7.56188$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16 − 16i)2-s − 512i·4-s + (2.97e3 − 946. i)5-s + (2.12e4 − 2.12e4i)7-s + (−8.19e3 − 8.19e3i)8-s + (3.25e4 − 6.27e4i)10-s − 1.55e5·11-s + (−3.58e5 − 3.58e5i)13-s − 6.81e5i·14-s − 2.62e5·16-s + (6.09e5 − 6.09e5i)17-s − 3.35e5i·19-s + (−4.84e5 − 1.52e6i)20-s + (−2.49e6 + 2.49e6i)22-s + (5.50e6 + 5.50e6i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.953 − 0.302i)5-s + (1.26 − 1.26i)7-s + (−0.250 − 0.250i)8-s + (0.325 − 0.627i)10-s − 0.966·11-s + (−0.965 − 0.965i)13-s − 1.26i·14-s − 0.250·16-s + (0.429 − 0.429i)17-s − 0.135i·19-s + (−0.151 − 0.476i)20-s + (−0.483 + 0.483i)22-s + (0.854 + 0.854i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.858 + 0.513i$
Analytic conductor: \(57.1821\)
Root analytic conductor: \(7.56188\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :5),\ -0.858 + 0.513i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.848406 - 3.06903i\)
\(L(\frac12)\) \(\approx\) \(0.848406 - 3.06903i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16 + 16i)T \)
3 \( 1 \)
5 \( 1 + (-2.97e3 + 946. i)T \)
good7 \( 1 + (-2.12e4 + 2.12e4i)T - 2.82e8iT^{2} \)
11 \( 1 + 1.55e5T + 2.59e10T^{2} \)
13 \( 1 + (3.58e5 + 3.58e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-6.09e5 + 6.09e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 3.35e5iT - 6.13e12T^{2} \)
23 \( 1 + (-5.50e6 - 5.50e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 1.33e6iT - 4.20e14T^{2} \)
31 \( 1 - 2.59e7T + 8.19e14T^{2} \)
37 \( 1 + (5.50e7 - 5.50e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.37e8T + 1.34e16T^{2} \)
43 \( 1 + (9.34e7 + 9.34e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.14e8 - 1.14e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-2.29e7 - 2.29e7i)T + 1.74e17iT^{2} \)
59 \( 1 + 8.73e8iT - 5.11e17T^{2} \)
61 \( 1 - 5.87e8T + 7.13e17T^{2} \)
67 \( 1 + (6.75e8 - 6.75e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 5.54e8T + 3.25e18T^{2} \)
73 \( 1 + (-8.91e8 - 8.91e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.69e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.96e9 + 1.96e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 7.73e9iT - 3.11e19T^{2} \)
97 \( 1 + (-1.16e10 + 1.16e10i)T - 7.37e19iT^{2} \)
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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

−11.60080109117518121299589240605, −10.46081945426552670519949489045, −9.937481605145253041452388624940, −8.225002366468511747680398373853, −7.10981931289313964930109873388, −5.26757988468421048129586398418, −4.84692437358308409910156652089, −3.05483193571533387068722187662, −1.70681687165062281966347109709, −0.63904970886948268362911485487, 1.85242984571175828718818326871, 2.70445553966943674742952857421, 4.82847353913611594152793472023, 5.44005108337235360358428335832, 6.69825170466009161954768444799, 8.051362367824416984168061739683, 9.047291507905375492679775457270, 10.39249721113840766575059103396, 11.66730184978634054012643616203, 12.56956170806569020166700232058

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