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Properties

Label 2-45-5.2-c8-0-12
Degree $2$
Conductor $45$
Sign $-0.933 + 0.359i$
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  + (−15.1 − 15.1i)2-s + 205. i·4-s + (−115. + 614. i)5-s + (605. + 605. i)7-s + (−767. + 767. i)8-s + (1.10e4 − 7.57e3i)10-s − 4.28e3·11-s + (2.89e4 − 2.89e4i)13-s − 1.84e4i·14-s + 7.59e4·16-s + (−4.99e4 − 4.99e4i)17-s + 1.26e4i·19-s + (−1.26e5 − 2.37e4i)20-s + (6.50e4 + 6.50e4i)22-s + (−1.04e5 + 1.04e5i)23-s + ⋯
L(s)  = 1  + (−0.949 − 0.949i)2-s + 0.802i·4-s + (−0.184 + 0.982i)5-s + (0.252 + 0.252i)7-s + (−0.187 + 0.187i)8-s + (1.10 − 0.757i)10-s − 0.292·11-s + (1.01 − 1.01i)13-s − 0.479i·14-s + 1.15·16-s + (−0.598 − 0.598i)17-s + 0.0968i·19-s + (−0.788 − 0.148i)20-s + (0.277 + 0.277i)22-s + (−0.374 + 0.374i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(18.3320\)
Root analytic conductor: \(4.28159\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :4),\ -0.933 + 0.359i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0944089 - 0.507935i\)
\(L(\frac12)\) \(\approx\) \(0.0944089 - 0.507935i\)
\(L(5)\) 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 \( 1 + (115. - 614. i)T \)
good2 \( 1 + (15.1 + 15.1i)T + 256iT^{2} \)
7 \( 1 + (-605. - 605. i)T + 5.76e6iT^{2} \)
11 \( 1 + 4.28e3T + 2.14e8T^{2} \)
13 \( 1 + (-2.89e4 + 2.89e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (4.99e4 + 4.99e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 1.26e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.04e5 - 1.04e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 6.40e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.30e6T + 8.52e11T^{2} \)
37 \( 1 + (-3.11e5 - 3.11e5i)T + 3.51e12iT^{2} \)
41 \( 1 + 3.20e6T + 7.98e12T^{2} \)
43 \( 1 + (-4.12e6 + 4.12e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-3.49e6 - 3.49e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-9.08e6 + 9.08e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 1.10e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.13e7T + 1.91e14T^{2} \)
67 \( 1 + (1.93e7 + 1.93e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 2.80e7T + 6.45e14T^{2} \)
73 \( 1 + (2.13e7 - 2.13e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 3.60e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.88e7 + 4.88e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 9.25e7iT - 3.93e15T^{2} \)
97 \( 1 + (-5.68e7 - 5.68e7i)T + 7.83e15iT^{2} \)
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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

−13.42821506062325259637388028006, −11.87083785939804472878095039260, −10.97531878002362078316265345873, −10.19663838328667125791715278687, −8.825863663346484716184403705994, −7.61660924932248633074791337915, −5.81206611089933538368912534465, −3.39692661387485347498152434703, −2.09283891912309544007323557696, −0.28499434691764942448845640968, 1.30857865654568395224872670575, 4.14065299941846020956535389945, 5.89848278499603003436055243819, 7.30021919325356564672069493049, 8.532253572035051528598058022376, 9.168136211566906832571233610206, 10.79074322357856075482142917340, 12.30818280399832906430228440479, 13.52792691763490660366833470932, 15.00098157251198535338639237170

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