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Properties

Label 2-45-5.4-c9-0-9
Degree $2$
Conductor $45$
Sign $-i$
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  + 42.4i·2-s − 1.29e3·4-s − 1.39e3i·5-s − 3.31e4i·8-s + 5.93e4·10-s + 7.47e5·16-s + 6.21e5i·17-s + 1.03e6·19-s + 1.80e6i·20-s − 2.63e6i·23-s − 1.95e6·25-s + 8.24e6·31-s + 1.47e7i·32-s − 2.63e7·34-s + 4.40e7i·38-s + ⋯
L(s)  = 1  + 1.87i·2-s − 2.52·4-s − 0.999i·5-s − 2.86i·8-s + 1.87·10-s + 2.85·16-s + 1.80i·17-s + 1.82·19-s + 2.52i·20-s − 1.96i·23-s − 1.00·25-s + 1.60·31-s + 2.49i·32-s − 3.38·34-s + 3.42i·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-i$
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: \((2,\ 45,\ (\ :9/2),\ -i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.08287 + 1.08287i\)
\(L(\frac12)\) \(\approx\) \(1.08287 + 1.08287i\)
\(L(\frac{11}{2})\) 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 \( 1 + 1.39e3iT \)
good2 \( 1 - 42.4iT - 512T^{2} \)
7 \( 1 - 4.03e7T^{2} \)
11 \( 1 + 2.35e9T^{2} \)
13 \( 1 - 1.06e10T^{2} \)
17 \( 1 - 6.21e5iT - 1.18e11T^{2} \)
19 \( 1 - 1.03e6T + 3.22e11T^{2} \)
23 \( 1 + 2.63e6iT - 1.80e12T^{2} \)
29 \( 1 + 1.45e13T^{2} \)
31 \( 1 - 8.24e6T + 2.64e13T^{2} \)
37 \( 1 - 1.29e14T^{2} \)
41 \( 1 + 3.27e14T^{2} \)
43 \( 1 - 5.02e14T^{2} \)
47 \( 1 - 7.52e6iT - 1.11e15T^{2} \)
53 \( 1 + 3.74e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.66e15T^{2} \)
61 \( 1 - 1.97e8T + 1.16e16T^{2} \)
67 \( 1 - 2.72e16T^{2} \)
71 \( 1 + 4.58e16T^{2} \)
73 \( 1 - 5.88e16T^{2} \)
79 \( 1 - 4.21e8T + 1.19e17T^{2} \)
83 \( 1 - 1.29e8iT - 1.86e17T^{2} \)
89 \( 1 + 3.50e17T^{2} \)
97 \( 1 - 7.60e17T^{2} \)
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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

−14.39050795843800414283777216891, −13.35270671581382327246873125176, −12.33035183801797398127775095437, −9.976163652145953607767638684842, −8.696523695355841272453904947583, −7.970322482951367425902250548765, −6.47271670271222045626871213742, −5.33143469526027423318825377743, −4.17393510414606820351914050895, −0.77654711725336274528612113687, 0.942253539110538920886575461622, 2.58945546741869791201188204583, 3.50519595016212747027509467680, 5.21346309866164254368266888697, 7.49329673863245663556279829675, 9.375004548382240436938630113086, 10.07634703402646714002057436171, 11.47151791461187282176768078253, 11.80380797241300411760486154128, 13.59131593919968040958953001643

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