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Properties

Label 2-90-15.2-c11-0-20
Degree $2$
Conductor $90$
Sign $-0.997 + 0.0676i$
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 + (4.26e3 − 5.53e3i)5-s + (−513. − 513. i)7-s + (−2.31e4 − 2.31e4i)8-s + (−2.89e4 − 2.21e5i)10-s − 3.06e5i·11-s + (5.45e5 − 5.45e5i)13-s − 2.32e4·14-s − 1.04e6·16-s + (−1.83e6 + 1.83e6i)17-s − 8.44e6i·19-s + (−5.67e6 − 4.36e6i)20-s + (−6.94e6 − 6.94e6i)22-s + (−6.12e6 − 6.12e6i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.609 − 0.792i)5-s + (−0.0115 − 0.0115i)7-s + (−0.250 − 0.250i)8-s + (−0.0913 − 0.701i)10-s − 0.574i·11-s + (0.407 − 0.407i)13-s − 0.0115·14-s − 0.250·16-s + (−0.312 + 0.312i)17-s − 0.782i·19-s + (−0.396 − 0.304i)20-s + (−0.287 − 0.287i)22-s + (−0.198 − 0.198i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0676i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.997 + 0.0676i$
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: \((2,\ 90,\ (\ :11/2),\ -0.997 + 0.0676i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0776768 - 2.29350i\)
\(L(\frac12)\) \(\approx\) \(0.0776768 - 2.29350i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-4.26e3 + 5.53e3i)T \)
good7 \( 1 + (513. + 513. i)T + 1.97e9iT^{2} \)
11 \( 1 + 3.06e5iT - 2.85e11T^{2} \)
13 \( 1 + (-5.45e5 + 5.45e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (1.83e6 - 1.83e6i)T - 3.42e13iT^{2} \)
19 \( 1 + 8.44e6iT - 1.16e14T^{2} \)
23 \( 1 + (6.12e6 + 6.12e6i)T + 9.52e14iT^{2} \)
29 \( 1 - 5.16e7T + 1.22e16T^{2} \)
31 \( 1 + 1.36e7T + 2.54e16T^{2} \)
37 \( 1 + (-5.48e7 - 5.48e7i)T + 1.77e17iT^{2} \)
41 \( 1 - 3.12e8iT - 5.50e17T^{2} \)
43 \( 1 + (-4.31e8 + 4.31e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (1.02e9 - 1.02e9i)T - 2.47e18iT^{2} \)
53 \( 1 + (1.45e9 + 1.45e9i)T + 9.26e18iT^{2} \)
59 \( 1 + 7.00e9T + 3.01e19T^{2} \)
61 \( 1 + 1.02e8T + 4.35e19T^{2} \)
67 \( 1 + (2.12e9 + 2.12e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 2.03e10iT - 2.31e20T^{2} \)
73 \( 1 + (1.99e10 - 1.99e10i)T - 3.13e20iT^{2} \)
79 \( 1 - 4.05e10iT - 7.47e20T^{2} \)
83 \( 1 + (2.14e10 + 2.14e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 5.80e10T + 2.77e21T^{2} \)
97 \( 1 + (-1.16e10 - 1.16e10i)T + 7.15e21iT^{2} \)
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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

−11.42370558301546488129892959526, −10.39361498375087873476723699380, −9.260225417143579941736933326895, −8.265381434286021208346121652599, −6.45632769262555195263689577747, −5.43802345132005035421710438996, −4.33541001066235241991724771821, −2.90059294513442639233283421193, −1.56625356213716367655963035216, −0.43564126814726060273704408728, 1.72282969837422948953106776415, 3.00751548109029097708692222689, 4.34047049567427053704146206664, 5.73971310575906579381319706195, 6.65777706308486715956610976956, 7.68714090563858680316310856440, 9.139554501954443586913564789079, 10.25916064117721292677546171640, 11.40970863456930342386851913623, 12.57105377344508448798887881419

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