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Properties

Label 2-45-5.4-c9-0-17
Degree $2$
Conductor $45$
Sign $-0.818 + 0.574i$
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  − 12.2i·2-s + 362.·4-s + (−1.14e3 + 803. i)5-s + 4.34e3i·7-s − 1.06e4i·8-s + (9.81e3 + 1.39e4i)10-s − 3.67e4·11-s − 1.87e5i·13-s + 5.30e4·14-s + 5.50e4·16-s + 3.94e3i·17-s + 2.37e5·19-s + (−4.14e5 + 2.91e5i)20-s + 4.49e5i·22-s − 2.19e6i·23-s + ⋯
L(s)  = 1  − 0.540i·2-s + 0.708·4-s + (−0.818 + 0.574i)5-s + 0.683i·7-s − 0.922i·8-s + (0.310 + 0.442i)10-s − 0.756·11-s − 1.82i·13-s + 0.369·14-s + 0.210·16-s + 0.0114i·17-s + 0.417·19-s + (−0.579 + 0.407i)20-s + 0.408i·22-s − 1.63i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.818 + 0.574i$
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: \((2,\ 45,\ (\ :9/2),\ -0.818 + 0.574i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.319226 - 1.01017i\)
\(L(\frac12)\) \(\approx\) \(0.319226 - 1.01017i\)
\(L(\frac{11}{2})\) 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 \( 1 + (1.14e3 - 803. i)T \)
good2 \( 1 + 12.2iT - 512T^{2} \)
7 \( 1 - 4.34e3iT - 4.03e7T^{2} \)
11 \( 1 + 3.67e4T + 2.35e9T^{2} \)
13 \( 1 + 1.87e5iT - 1.06e10T^{2} \)
17 \( 1 - 3.94e3iT - 1.18e11T^{2} \)
19 \( 1 - 2.37e5T + 3.22e11T^{2} \)
23 \( 1 + 2.19e6iT - 1.80e12T^{2} \)
29 \( 1 + 6.47e6T + 1.45e13T^{2} \)
31 \( 1 + 4.92e6T + 2.64e13T^{2} \)
37 \( 1 - 6.61e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.22e7T + 3.27e14T^{2} \)
43 \( 1 + 1.25e7iT - 5.02e14T^{2} \)
47 \( 1 + 3.42e7iT - 1.11e15T^{2} \)
53 \( 1 - 3.09e7iT - 3.29e15T^{2} \)
59 \( 1 - 7.50e7T + 8.66e15T^{2} \)
61 \( 1 - 1.30e7T + 1.16e16T^{2} \)
67 \( 1 - 1.37e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.12e8T + 4.58e16T^{2} \)
73 \( 1 + 2.65e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.69e8T + 1.19e17T^{2} \)
83 \( 1 - 5.22e8iT - 1.86e17T^{2} \)
89 \( 1 - 3.29e8T + 3.50e17T^{2} \)
97 \( 1 + 1.12e9iT - 7.60e17T^{2} \)
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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

−12.97246813478117923553165435206, −12.12683544720310304017592701599, −10.97357946189056503730462691262, −10.21913993296446254188884956554, −8.262664387501124929008862411311, −7.15019873206511938592225170376, −5.56880080779020026025246864216, −3.43043298591612643097573338262, −2.43019326337793910592443227917, −0.33965086271740723864416505606, 1.68107266329201587971483229568, 3.76876722703005229222261280469, 5.32508528193697811554590446978, 7.04627361071236380142010749380, 7.76993717119441715558051805867, 9.287779131646571957431606092515, 11.06824266834025094016153130614, 11.76624323689540720070048311848, 13.25056169706402218679284890861, 14.51572390146767509052226174113

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