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Properties

Label 2-45-9.5-c4-0-4
Degree $2$
Conductor $45$
Sign $0.227 - 0.973i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.817 − 0.471i)2-s + (−6.73 − 5.97i)3-s + (−7.55 + 13.0i)4-s + (9.68 + 5.59i)5-s + (−8.31 − 1.70i)6-s + (31.6 + 54.8i)7-s + 29.3i·8-s + (9.62 + 80.4i)9-s + 10.5·10-s + (−1.86 + 1.07i)11-s + (129. − 42.9i)12-s + (−49.0 + 84.9i)13-s + (51.7 + 29.9i)14-s + (−31.7 − 95.4i)15-s + (−107. − 185. i)16-s + 193. i·17-s + ⋯
L(s)  = 1  + (0.204 − 0.117i)2-s + (−0.747 − 0.663i)3-s + (−0.472 + 0.817i)4-s + (0.387 + 0.223i)5-s + (−0.231 − 0.0473i)6-s + (0.646 + 1.12i)7-s + 0.458i·8-s + (0.118 + 0.992i)9-s + 0.105·10-s + (−0.0153 + 0.00888i)11-s + (0.896 − 0.298i)12-s + (−0.290 + 0.502i)13-s + (0.264 + 0.152i)14-s + (−0.141 − 0.424i)15-s + (−0.418 − 0.724i)16-s + 0.670i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.227 - 0.973i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.227 - 0.973i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.871599 + 0.691101i\)
\(L(\frac12)\) \(\approx\) \(0.871599 + 0.691101i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.73 + 5.97i)T \)
5 \( 1 + (-9.68 - 5.59i)T \)
good2 \( 1 + (-0.817 + 0.471i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (-31.6 - 54.8i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (1.86 - 1.07i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (49.0 - 84.9i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 193. iT - 8.35e4T^{2} \)
19 \( 1 + 473.T + 1.30e5T^{2} \)
23 \( 1 + (-481. - 277. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-372. + 215. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-556. + 964. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.63e3T + 1.87e6T^{2} \)
41 \( 1 + (-2.10e3 - 1.21e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-766. - 1.32e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.18e3 + 1.25e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 4.49e3iT - 7.89e6T^{2} \)
59 \( 1 + (4.01e3 + 2.31e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (26.4 + 45.8i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.80e3 + 3.12e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 9.59e3iT - 2.54e7T^{2} \)
73 \( 1 + 302.T + 2.83e7T^{2} \)
79 \( 1 + (-5.03e3 - 8.72e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-7.73e3 + 4.46e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 8.73e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.80e3 - 8.32e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.23506163571931100245862382655, −13.91801583309583978930104617896, −12.78639546882017053437784717012, −12.01297990614016685780602912277, −10.93711567137049780459391300597, −9.000573922374484528585980800589, −7.79499155326254626541488887123, −6.15800216108475822445549736362, −4.73952077015653037257743412299, −2.25603717987245092737310136953, 0.76206046771197851456622261693, 4.34298992682278763369675082046, 5.28692347019650117108409764196, 6.79222142261545363464265050934, 8.927025099425010279595998704213, 10.31400036441891896493127237938, 10.80905434446253492942988518954, 12.56729160043592275922067417881, 13.89345409741689304574049551249, 14.77154503044725137604123435510

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