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Properties

Label 2-45-5.4-c9-0-21
Degree $2$
Conductor $45$
Sign $-0.540 - 0.841i$
Analytic cond. $23.1766$
Root an. cond. $4.81420$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 30.1i·2-s − 398.·4-s + (−755. − 1.17e3i)5-s − 1.02e4i·7-s − 3.42e3i·8-s + (−3.54e4 + 2.27e4i)10-s + 9.07e4·11-s − 1.14e5i·13-s − 3.09e5·14-s − 3.07e5·16-s + 2.07e5i·17-s − 7.57e5·19-s + (3.01e5 + 4.68e5i)20-s − 2.73e6i·22-s + 9.13e5i·23-s + ⋯
L(s)  = 1  − 1.33i·2-s − 0.778·4-s + (−0.540 − 0.841i)5-s − 1.61i·7-s − 0.295i·8-s + (−1.12 + 0.720i)10-s + 1.86·11-s − 1.11i·13-s − 2.15·14-s − 1.17·16-s + 0.603i·17-s − 1.33·19-s + (0.420 + 0.655i)20-s − 2.49i·22-s + 0.680i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(23.1766\)
Root analytic conductor: \(4.81420\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :9/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.754792 + 1.38191i\)
\(L(\frac12)\) \(\approx\) \(0.754792 + 1.38191i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (755. + 1.17e3i)T \)
good2 \( 1 + 30.1iT - 512T^{2} \)
7 \( 1 + 1.02e4iT - 4.03e7T^{2} \)
11 \( 1 - 9.07e4T + 2.35e9T^{2} \)
13 \( 1 + 1.14e5iT - 1.06e10T^{2} \)
17 \( 1 - 2.07e5iT - 1.18e11T^{2} \)
19 \( 1 + 7.57e5T + 3.22e11T^{2} \)
23 \( 1 - 9.13e5iT - 1.80e12T^{2} \)
29 \( 1 - 7.82e5T + 1.45e13T^{2} \)
31 \( 1 - 7.94e6T + 2.64e13T^{2} \)
37 \( 1 - 7.25e6iT - 1.29e14T^{2} \)
41 \( 1 - 5.80e4T + 3.27e14T^{2} \)
43 \( 1 + 1.73e7iT - 5.02e14T^{2} \)
47 \( 1 + 2.02e7iT - 1.11e15T^{2} \)
53 \( 1 - 4.71e6iT - 3.29e15T^{2} \)
59 \( 1 - 9.45e7T + 8.66e15T^{2} \)
61 \( 1 + 5.44e7T + 1.16e16T^{2} \)
67 \( 1 + 1.52e8iT - 2.72e16T^{2} \)
71 \( 1 - 3.39e8T + 4.58e16T^{2} \)
73 \( 1 - 2.34e8iT - 5.88e16T^{2} \)
79 \( 1 + 4.97e6T + 1.19e17T^{2} \)
83 \( 1 - 1.74e8iT - 1.86e17T^{2} \)
89 \( 1 + 6.42e8T + 3.50e17T^{2} \)
97 \( 1 + 4.54e8iT - 7.60e17T^{2} \)
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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

−12.81884121771486838463123228598, −11.85445003447717057422894961636, −10.79812492315173055429716480443, −9.794654572409611290653912115725, −8.359449044347774362930128982708, −6.75116455664065575934296379351, −4.31140635651742738156895706066, −3.64632153983978601035844239487, −1.38874532077118551801707861388, −0.58801489903530422885681153116, 2.35608959522620515023841156039, 4.39606919536586108834765337756, 6.24425887531194310329137424844, 6.72108173973090648573801282072, 8.380824069198958270789311362832, 9.238846907971189214548425134475, 11.38826872212567852761771440589, 12.08263699095345214764194505288, 14.21852220689517025015152080975, 14.72922649197242406114831461664

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