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Properties

Label 2-45-5.3-c8-0-14
Degree $2$
Conductor $45$
Sign $0.246 + 0.969i$
Analytic cond. $18.3320$
Root an. cond. $4.28159$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.8 − 11.8i)2-s − 25.5i·4-s + (434. − 449. i)5-s + (508. − 508. i)7-s + (2.73e3 + 2.73e3i)8-s + (−178. − 1.04e4i)10-s + 7.02e3·11-s + (−8.07e3 − 8.07e3i)13-s − 1.20e4i·14-s + 7.14e4·16-s + (1.02e5 − 1.02e5i)17-s − 5.95e4i·19-s + (−1.14e4 − 1.10e4i)20-s + (8.33e4 − 8.33e4i)22-s + (−1.32e5 − 1.32e5i)23-s + ⋯
L(s)  = 1  + (0.741 − 0.741i)2-s − 0.0998i·4-s + (0.694 − 0.719i)5-s + (0.211 − 0.211i)7-s + (0.667 + 0.667i)8-s + (−0.0178 − 1.04i)10-s + 0.479·11-s + (−0.282 − 0.282i)13-s − 0.313i·14-s + 1.08·16-s + (1.23 − 1.23i)17-s − 0.457i·19-s + (−0.0717 − 0.0693i)20-s + (0.355 − 0.355i)22-s + (−0.474 − 0.474i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(18.3320\)
Root analytic conductor: \(4.28159\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :4),\ 0.246 + 0.969i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.67153 - 2.07765i\)
\(L(\frac12)\) \(\approx\) \(2.67153 - 2.07765i\)
\(L(5)\) 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 \( 1 + (-434. + 449. i)T \)
good2 \( 1 + (-11.8 + 11.8i)T - 256iT^{2} \)
7 \( 1 + (-508. + 508. i)T - 5.76e6iT^{2} \)
11 \( 1 - 7.02e3T + 2.14e8T^{2} \)
13 \( 1 + (8.07e3 + 8.07e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (-1.02e5 + 1.02e5i)T - 6.97e9iT^{2} \)
19 \( 1 + 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.32e5 + 1.32e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 3.92e5iT - 5.00e11T^{2} \)
31 \( 1 + 5.07e5T + 8.52e11T^{2} \)
37 \( 1 + (6.10e4 - 6.10e4i)T - 3.51e12iT^{2} \)
41 \( 1 - 1.81e6T + 7.98e12T^{2} \)
43 \( 1 + (-1.47e6 - 1.47e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-1.79e6 + 1.79e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-5.66e6 - 5.66e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.74e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.96e7T + 1.91e14T^{2} \)
67 \( 1 + (1.12e7 - 1.12e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 3.01e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.52e7 - 2.52e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 8.14e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.99e7 - 1.99e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 8.20e7iT - 3.93e15T^{2} \)
97 \( 1 + (-3.37e7 + 3.37e7i)T - 7.83e15iT^{2} \)
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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

−13.71201280271741698294829733837, −12.60663564113799666228187119079, −11.81582186664489453152329375684, −10.42790241041124606200527646798, −9.102832530272418155263178163762, −7.56798144475209029646761134830, −5.54465065492804848449689145104, −4.39879984731979531386814861837, −2.73843146706331489213997769811, −1.17055407090526799329798797296, 1.68034801586190557266335142379, 3.76864501468675446529485174879, 5.50262297780090319601002982099, 6.36200958481848116998214941467, 7.69089375178239599415481514447, 9.613102370318944349705354261264, 10.64170818907115875426144583340, 12.26018037281379722208045842182, 13.58290872979492907422684240252, 14.49023057482168540525754719521

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