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Properties

Label 2-90-5.4-c5-0-8
Degree $2$
Conductor $90$
Sign $0.178 + 0.983i$
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 + (55 − 10i)5-s − 4i·7-s + 64i·8-s + (−40 − 220i)10-s + 500·11-s + 288i·13-s − 16·14-s + 256·16-s − 1.51e3i·17-s + 1.34e3·19-s + (−880 + 160i)20-s − 2.00e3i·22-s − 4.10e3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.983 − 0.178i)5-s − 0.0308i·7-s + 0.353i·8-s + (−0.126 − 0.695i)10-s + 1.24·11-s + 0.472i·13-s − 0.0218·14-s + 0.250·16-s − 1.27i·17-s + 0.854·19-s + (−0.491 + 0.0894i)20-s − 0.880i·22-s − 1.61i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.178 + 0.983i$
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: \((2,\ 90,\ (\ :5/2),\ 0.178 + 0.983i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.63276 - 1.36267i\)
\(L(\frac12)\) \(\approx\) \(1.63276 - 1.36267i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55 + 10i)T \)
good7 \( 1 + 4iT - 1.68e4T^{2} \)
11 \( 1 - 500T + 1.61e5T^{2} \)
13 \( 1 - 288iT - 3.71e5T^{2} \)
17 \( 1 + 1.51e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.34e3T + 2.47e6T^{2} \)
23 \( 1 + 4.10e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.64e3T + 2.05e7T^{2} \)
31 \( 1 + 5.61e3T + 2.86e7T^{2} \)
37 \( 1 + 7.28e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.89e4T + 1.15e8T^{2} \)
43 \( 1 - 2.40e3iT - 1.47e8T^{2} \)
47 \( 1 + 8.90e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.98e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.83e4T + 7.14e8T^{2} \)
61 \( 1 - 1.82e4T + 8.44e8T^{2} \)
67 \( 1 - 6.59e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.88e4T + 1.80e9T^{2} \)
73 \( 1 - 3.08e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.02e4T + 3.07e9T^{2} \)
83 \( 1 + 2.46e3iT - 3.93e9T^{2} \)
89 \( 1 - 2.26e4T + 5.58e9T^{2} \)
97 \( 1 + 3.69e4iT - 8.58e9T^{2} \)
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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

−12.81870899589157661423806784592, −11.85336586988649920030853644940, −10.77556339175268371557499246693, −9.477751498930544868759119109792, −9.005040534915891400101713237691, −7.08438496006970519398732240640, −5.68668026225452014553878987905, −4.25716412788135121132275490878, −2.50475634595175616871834590523, −1.02735312446622005614696693660, 1.48304210697544942514375858582, 3.64115891137828773068691009250, 5.43232744380374293338938650887, 6.32560384392382691006213829588, 7.58352100319190299151170309654, 9.040172510451602001855129483472, 9.800657460147906016716225429453, 11.15086976163081384741291349722, 12.59253517828722496480906700930, 13.59008462153963374369562770331

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