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Properties

Label 2-90-15.2-c11-0-10
Degree $2$
Conductor $90$
Sign $0.941 - 0.338i$
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 + (2.58e3 − 6.49e3i)5-s + (−1.39e4 − 1.39e4i)7-s + (2.31e4 + 2.31e4i)8-s + (8.84e4 + 2.05e5i)10-s − 9.03e4i·11-s + (−1.75e5 + 1.75e5i)13-s + 6.29e5·14-s − 1.04e6·16-s + (−3.11e6 + 3.11e6i)17-s + 1.81e7i·19-s + (−6.64e6 − 2.64e6i)20-s + (2.04e6 + 2.04e6i)22-s + (3.12e7 + 3.12e7i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.369 − 0.929i)5-s + (−0.312 − 0.312i)7-s + (0.250 + 0.250i)8-s + (0.279 + 0.649i)10-s − 0.169i·11-s + (−0.131 + 0.131i)13-s + 0.312·14-s − 0.250·16-s + (−0.532 + 0.532i)17-s + 1.67i·19-s + (−0.464 − 0.184i)20-s + (0.0845 + 0.0845i)22-s + (1.01 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.941 - 0.338i$
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: \((2,\ 90,\ (\ :11/2),\ 0.941 - 0.338i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.46151 + 0.254653i\)
\(L(\frac12)\) \(\approx\) \(1.46151 + 0.254653i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-2.58e3 + 6.49e3i)T \)
good7 \( 1 + (1.39e4 + 1.39e4i)T + 1.97e9iT^{2} \)
11 \( 1 + 9.03e4iT - 2.85e11T^{2} \)
13 \( 1 + (1.75e5 - 1.75e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (3.11e6 - 3.11e6i)T - 3.42e13iT^{2} \)
19 \( 1 - 1.81e7iT - 1.16e14T^{2} \)
23 \( 1 + (-3.12e7 - 3.12e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 9.60e7T + 1.22e16T^{2} \)
31 \( 1 - 2.45e8T + 2.54e16T^{2} \)
37 \( 1 + (2.51e8 + 2.51e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 1.94e8iT - 5.50e17T^{2} \)
43 \( 1 + (-9.78e8 + 9.78e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (1.90e8 - 1.90e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (-1.32e9 - 1.32e9i)T + 9.26e18iT^{2} \)
59 \( 1 + 2.40e9T + 3.01e19T^{2} \)
61 \( 1 - 5.09e9T + 4.35e19T^{2} \)
67 \( 1 + (2.14e9 + 2.14e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 5.51e9iT - 2.31e20T^{2} \)
73 \( 1 + (-1.21e10 + 1.21e10i)T - 3.13e20iT^{2} \)
79 \( 1 + 8.48e9iT - 7.47e20T^{2} \)
83 \( 1 + (-3.27e10 - 3.27e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 6.12e10T + 2.77e21T^{2} \)
97 \( 1 + (-5.81e10 - 5.81e10i)T + 7.15e21iT^{2} \)
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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

−12.00476472458784078133047161531, −10.57703513544341423171199343471, −9.594929849730722929738084409955, −8.673148283269035683086566839098, −7.61318155058743144641571249516, −6.26111723236645301456185266324, −5.25589979430739950578955045428, −3.85250036546631013479590491273, −1.87274922023457015415363220079, −0.75249159061253992869536016380, 0.62587778509633549909223776298, 2.33221827916875597122001801623, 3.02760821819497700224534348760, 4.74165486819007060590097003326, 6.42105126819540363490550406614, 7.28937788425515343909524173109, 8.810159859487670106896173983752, 9.698058334222232648486660985606, 10.76910020630535748586669941815, 11.53479905175960281930646126389

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