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Properties

Label 2-90-15.2-c11-0-13
Degree $2$
Conductor $90$
Sign $0.932 + 0.362i$
Analytic cond. $69.1508$
Root an. cond. $8.31570$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (22.6 − 22.6i)2-s − 1.02e3i·4-s + (5.09e3 + 4.78e3i)5-s + (2.25e4 + 2.25e4i)7-s + (−2.31e4 − 2.31e4i)8-s + (2.23e5 − 6.95e3i)10-s − 3.72e5i·11-s + (3.53e4 − 3.53e4i)13-s + 1.02e6·14-s − 1.04e6·16-s + (4.44e5 − 4.44e5i)17-s − 1.44e7i·19-s + (4.89e6 − 5.21e6i)20-s + (−8.42e6 − 8.42e6i)22-s + (3.55e7 + 3.55e7i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.728 + 0.684i)5-s + (0.506 + 0.506i)7-s + (−0.250 − 0.250i)8-s + (0.706 − 0.0219i)10-s − 0.697i·11-s + (0.0263 − 0.0263i)13-s + 0.506·14-s − 0.250·16-s + (0.0758 − 0.0758i)17-s − 1.33i·19-s + (0.342 − 0.364i)20-s + (−0.348 − 0.348i)22-s + (1.15 + 1.15i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.362i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.932 + 0.362i$
Analytic conductor: \(69.1508\)
Root analytic conductor: \(8.31570\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :11/2),\ 0.932 + 0.362i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.61103 - 0.676989i\)
\(L(\frac12)\) \(\approx\) \(3.61103 - 0.676989i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-22.6 + 22.6i)T \)
3 \( 1 \)
5 \( 1 + (-5.09e3 - 4.78e3i)T \)
good7 \( 1 + (-2.25e4 - 2.25e4i)T + 1.97e9iT^{2} \)
11 \( 1 + 3.72e5iT - 2.85e11T^{2} \)
13 \( 1 + (-3.53e4 + 3.53e4i)T - 1.79e12iT^{2} \)
17 \( 1 + (-4.44e5 + 4.44e5i)T - 3.42e13iT^{2} \)
19 \( 1 + 1.44e7iT - 1.16e14T^{2} \)
23 \( 1 + (-3.55e7 - 3.55e7i)T + 9.52e14iT^{2} \)
29 \( 1 - 1.37e7T + 1.22e16T^{2} \)
31 \( 1 + 3.19e7T + 2.54e16T^{2} \)
37 \( 1 + (-5.18e8 - 5.18e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 3.08e8iT - 5.50e17T^{2} \)
43 \( 1 + (-3.54e8 + 3.54e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (-6.15e8 + 6.15e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (-1.29e9 - 1.29e9i)T + 9.26e18iT^{2} \)
59 \( 1 - 2.69e9T + 3.01e19T^{2} \)
61 \( 1 - 1.34e9T + 4.35e19T^{2} \)
67 \( 1 + (-1.39e10 - 1.39e10i)T + 1.22e20iT^{2} \)
71 \( 1 + 2.01e10iT - 2.31e20T^{2} \)
73 \( 1 + (-2.99e9 + 2.99e9i)T - 3.13e20iT^{2} \)
79 \( 1 - 2.00e10iT - 7.47e20T^{2} \)
83 \( 1 + (3.15e10 + 3.15e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 3.82e10T + 2.77e21T^{2} \)
97 \( 1 + (-3.73e10 - 3.73e10i)T + 7.15e21iT^{2} \)
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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

−11.55889415404113123515713225488, −11.04291016171648048563229129582, −9.793160097755719272904109412876, −8.759875934200898066497186346367, −7.10984041587607391990407649224, −5.89602278441699038486258034817, −4.94430779140133146896565829149, −3.25726529444713226637226671386, −2.33256583093396896680643543878, −1.00347965857305393224277708034, 0.951114537548054586393576564039, 2.26438297069066418619441167611, 4.06772156624728898691908772325, 5.01803884326947132126327024398, 6.12905068752235783828721481921, 7.40717013734322661864199665066, 8.508497573523027439617959163118, 9.697067702364389040492642374854, 10.87668618475251515957001380508, 12.36074151095103603063742950526

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