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Properties

Label 2-45-5.4-c9-0-11
Degree $2$
Conductor $45$
Sign $0.814 - 0.580i$
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  + 41.3i·2-s − 1.19e3·4-s + (−1.13e3 + 810. i)5-s − 5.31e3i·7-s − 2.82e4i·8-s + (−3.35e4 − 4.70e4i)10-s − 1.04e4·11-s + 7.96e4i·13-s + 2.19e5·14-s + 5.54e5·16-s − 3.13e5i·17-s + 2.46e5·19-s + (1.36e6 − 9.69e5i)20-s − 4.30e5i·22-s + 7.21e5i·23-s + ⋯
L(s)  = 1  + 1.82i·2-s − 2.33·4-s + (−0.814 + 0.580i)5-s − 0.836i·7-s − 2.43i·8-s + (−1.05 − 1.48i)10-s − 0.214·11-s + 0.773i·13-s + 1.52·14-s + 2.11·16-s − 0.911i·17-s + 0.434·19-s + (1.90 − 1.35i)20-s − 0.392i·22-s + 0.537i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.814 - 0.580i$
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: \((2,\ 45,\ (\ :9/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.693337 + 0.221690i\)
\(L(\frac12)\) \(\approx\) \(0.693337 + 0.221690i\)
\(L(\frac{11}{2})\) 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 \( 1 + (1.13e3 - 810. i)T \)
good2 \( 1 - 41.3iT - 512T^{2} \)
7 \( 1 + 5.31e3iT - 4.03e7T^{2} \)
11 \( 1 + 1.04e4T + 2.35e9T^{2} \)
13 \( 1 - 7.96e4iT - 1.06e10T^{2} \)
17 \( 1 + 3.13e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.46e5T + 3.22e11T^{2} \)
23 \( 1 - 7.21e5iT - 1.80e12T^{2} \)
29 \( 1 - 2.56e6T + 1.45e13T^{2} \)
31 \( 1 + 3.29e6T + 2.64e13T^{2} \)
37 \( 1 - 1.40e7iT - 1.29e14T^{2} \)
41 \( 1 + 1.70e7T + 3.27e14T^{2} \)
43 \( 1 + 2.92e7iT - 5.02e14T^{2} \)
47 \( 1 + 4.10e7iT - 1.11e15T^{2} \)
53 \( 1 + 5.67e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.60e8T + 8.66e15T^{2} \)
61 \( 1 - 5.33e7T + 1.16e16T^{2} \)
67 \( 1 + 2.80e8iT - 2.72e16T^{2} \)
71 \( 1 - 8.97e7T + 4.58e16T^{2} \)
73 \( 1 - 7.60e7iT - 5.88e16T^{2} \)
79 \( 1 + 4.10e8T + 1.19e17T^{2} \)
83 \( 1 + 5.21e8iT - 1.86e17T^{2} \)
89 \( 1 - 2.37e8T + 3.50e17T^{2} \)
97 \( 1 + 6.03e8iT - 7.60e17T^{2} \)
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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

−14.21320077591780450413863573224, −13.49375584305598019702498192902, −11.67659274485166600315257485373, −10.01683502726741744639455416897, −8.540856198010319033956457532295, −7.35187250360813660435780509612, −6.76839267446191538850381642785, −5.03399792452507273727433373013, −3.74729646524425962088778689553, −0.31429343395810081112735167126, 1.07176890370784029230433237863, 2.65893894080984755902079553637, 3.97000821875881481621849033512, 5.32790441163938011720935577037, 8.138615209343430532338855799889, 9.084378023060380496369935654272, 10.41852288928924480766389305093, 11.49289657306017414164273980768, 12.44327428851938490376377969649, 13.01880040682947904977920914897

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