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Properties

Label 2-90-5.4-c5-0-3
Degree $2$
Conductor $90$
Sign $0.939 - 0.342i$
Analytic cond. $14.4345$
Root an. cond. $3.79928$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 16·4-s + (−19.1 − 52.5i)5-s + 233. i·7-s + 64i·8-s + (−210. + 76.6i)10-s + 89.7·11-s + 209. i·13-s + 934.·14-s + 256·16-s − 226. i·17-s + 2.18e3·19-s + (306. + 840. i)20-s − 358. i·22-s + 4.29e3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.342 − 0.939i)5-s + 1.80i·7-s + 0.353i·8-s + (−0.664 + 0.242i)10-s + 0.223·11-s + 0.343i·13-s + 1.27·14-s + 0.250·16-s − 0.190i·17-s + 1.38·19-s + (0.171 + 0.469i)20-s − 0.158i·22-s + 1.69i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(14.4345\)
Root analytic conductor: \(3.79928\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :5/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.36628 + 0.241598i\)
\(L(\frac12)\) \(\approx\) \(1.36628 + 0.241598i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 \)
5 \( 1 + (19.1 + 52.5i)T \)
good7 \( 1 - 233. iT - 1.68e4T^{2} \)
11 \( 1 - 89.7T + 1.61e5T^{2} \)
13 \( 1 - 209. iT - 3.71e5T^{2} \)
17 \( 1 + 226. iT - 1.41e6T^{2} \)
19 \( 1 - 2.18e3T + 2.47e6T^{2} \)
23 \( 1 - 4.29e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.55e3T + 2.05e7T^{2} \)
31 \( 1 - 6.90e3T + 2.86e7T^{2} \)
37 \( 1 + 5.43e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.48e3T + 1.15e8T^{2} \)
43 \( 1 - 2.19e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.71e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.27e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.45e4T + 7.14e8T^{2} \)
61 \( 1 - 1.15e4T + 8.44e8T^{2} \)
67 \( 1 + 3.96e3iT - 1.35e9T^{2} \)
71 \( 1 + 4.64e4T + 1.80e9T^{2} \)
73 \( 1 - 6.15e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.78e4T + 3.07e9T^{2} \)
83 \( 1 + 5.09e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.13e3T + 5.58e9T^{2} \)
97 \( 1 + 8.84e4iT - 8.58e9T^{2} \)
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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

−12.91755360760861010767893372305, −11.85658782447411997907933163197, −11.64136087934815188281093069459, −9.622120263223117780576175034245, −9.038074083933238033714885708084, −7.918717843469619865282069881171, −5.81208260368957187526211831296, −4.78124084389063225300119654268, −3.02865817428955764013586754441, −1.40345132069366578138288248326, 0.63555381370487539843732611041, 3.35345012874473822826006611882, 4.57938670342587093584989947092, 6.47980915617240517533315970075, 7.23774463745202411789122188585, 8.220209493557013807454592707931, 10.04120535284229803363770340673, 10.62663350560730965788993198614, 12.01453544248018575629619485269, 13.61106067747413586660943377649

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