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Properties

Label 2-90-15.2-c11-0-18
Degree $2$
Conductor $90$
Sign $0.487 + 0.873i$
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 + (6.82e3 + 1.48e3i)5-s + (2.82e4 + 2.82e4i)7-s + (2.31e4 + 2.31e4i)8-s + (−1.88e5 + 1.20e5i)10-s − 7.89e5i·11-s + (1.06e6 − 1.06e6i)13-s − 1.27e6·14-s − 1.04e6·16-s + (1.64e4 − 1.64e4i)17-s − 1.98e7i·19-s + (1.52e6 − 6.99e6i)20-s + (1.78e7 + 1.78e7i)22-s + (1.44e7 + 1.44e7i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.977 + 0.213i)5-s + (0.635 + 0.635i)7-s + (0.250 + 0.250i)8-s + (−0.595 + 0.381i)10-s − 1.47i·11-s + (0.796 − 0.796i)13-s − 0.635·14-s − 0.250·16-s + (0.00281 − 0.00281i)17-s − 1.84i·19-s + (0.106 − 0.488i)20-s + (0.738 + 0.738i)22-s + (0.467 + 0.467i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.487 + 0.873i$
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: \((2,\ 90,\ (\ :11/2),\ 0.487 + 0.873i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.51041 - 0.887135i\)
\(L(\frac12)\) \(\approx\) \(1.51041 - 0.887135i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-6.82e3 - 1.48e3i)T \)
good7 \( 1 + (-2.82e4 - 2.82e4i)T + 1.97e9iT^{2} \)
11 \( 1 + 7.89e5iT - 2.85e11T^{2} \)
13 \( 1 + (-1.06e6 + 1.06e6i)T - 1.79e12iT^{2} \)
17 \( 1 + (-1.64e4 + 1.64e4i)T - 3.42e13iT^{2} \)
19 \( 1 + 1.98e7iT - 1.16e14T^{2} \)
23 \( 1 + (-1.44e7 - 1.44e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 1.16e8T + 1.22e16T^{2} \)
31 \( 1 + 2.16e8T + 2.54e16T^{2} \)
37 \( 1 + (4.63e8 + 4.63e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 9.64e8iT - 5.50e17T^{2} \)
43 \( 1 + (9.18e8 - 9.18e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (6.92e7 - 6.92e7i)T - 2.47e18iT^{2} \)
53 \( 1 + (-1.91e9 - 1.91e9i)T + 9.26e18iT^{2} \)
59 \( 1 + 2.94e9T + 3.01e19T^{2} \)
61 \( 1 + 8.64e9T + 4.35e19T^{2} \)
67 \( 1 + (-4.06e9 - 4.06e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.93e10iT - 2.31e20T^{2} \)
73 \( 1 + (1.06e9 - 1.06e9i)T - 3.13e20iT^{2} \)
79 \( 1 + 5.01e10iT - 7.47e20T^{2} \)
83 \( 1 + (-6.84e8 - 6.84e8i)T + 1.28e21iT^{2} \)
89 \( 1 - 7.45e10T + 2.77e21T^{2} \)
97 \( 1 + (6.86e10 + 6.86e10i)T + 7.15e21iT^{2} \)
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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

−11.24787085484347904216107448358, −10.72959286591888023292068535658, −9.135473738856679257806277827850, −8.665805159660883856347998713499, −7.21403394011198216641800994948, −5.86444752123483777799942889991, −5.30961125357535766773811933739, −3.15034535889835432199029870275, −1.77128958443303725657075587810, −0.48164213762763922781547930923, 1.46627102052731700207899930207, 1.85629011185162823199049800175, 3.77513587335979920278795792267, 5.00082112238032045034136282523, 6.56137464176009091195520324375, 7.75602164476102661313984000199, 9.005660234228775315457308493456, 9.968291180898999852758537104204, 10.75444481600512600450261974116, 12.02439586417494280263972085693

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