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Properties

Label 2-45-9.5-c4-0-7
Degree $2$
Conductor $45$
Sign $0.868 + 0.494i$
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  + (−3.23 + 1.86i)2-s + (−8.96 − 0.758i)3-s + (−1.01 + 1.76i)4-s + (−9.68 − 5.59i)5-s + (30.4 − 14.3i)6-s + (8.02 + 13.9i)7-s − 67.3i·8-s + (79.8 + 13.5i)9-s + 41.7·10-s + (134. − 77.8i)11-s + (10.4 − 15.0i)12-s + (−11.5 + 19.9i)13-s + (−51.9 − 30.0i)14-s + (82.5 + 57.4i)15-s + (109. + 189. i)16-s − 256. i·17-s + ⋯
L(s)  = 1  + (−0.809 + 0.467i)2-s + (−0.996 − 0.0842i)3-s + (−0.0635 + 0.110i)4-s + (−0.387 − 0.223i)5-s + (0.845 − 0.397i)6-s + (0.163 + 0.283i)7-s − 1.05i·8-s + (0.985 + 0.167i)9-s + 0.417·10-s + (1.11 − 0.643i)11-s + (0.0726 − 0.104i)12-s + (−0.0683 + 0.118i)13-s + (−0.265 − 0.153i)14-s + (0.367 + 0.255i)15-s + (0.428 + 0.741i)16-s − 0.887i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.494i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.868 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.868 + 0.494i$
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: \((2,\ 45,\ (\ :2),\ 0.868 + 0.494i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.532692 - 0.141069i\)
\(L(\frac12)\) \(\approx\) \(0.532692 - 0.141069i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.96 + 0.758i)T \)
5 \( 1 + (9.68 + 5.59i)T \)
good2 \( 1 + (3.23 - 1.86i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (-8.02 - 13.9i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-134. + 77.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (11.5 - 19.9i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 256. iT - 8.35e4T^{2} \)
19 \( 1 - 163.T + 1.30e5T^{2} \)
23 \( 1 + (707. + 408. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-493. + 284. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-223. + 386. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 2.11e3T + 1.87e6T^{2} \)
41 \( 1 + (-336. - 194. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.17e3 + 2.02e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-3.63e3 + 2.10e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.69e3iT - 7.89e6T^{2} \)
59 \( 1 + (4.68e3 + 2.70e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (733. + 1.27e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.11e3 - 5.39e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 2.71e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.00e3T + 2.83e7T^{2} \)
79 \( 1 + (4.25e3 + 7.37e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (5.54e3 - 3.20e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 6.12e3iT - 6.27e7T^{2} \)
97 \( 1 + (220. + 382. i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.54283611038363860877166532340, −13.79220493827334295578447277132, −12.28490245066299939703509281855, −11.56593135289698016310138547947, −9.957196577291624669174638889667, −8.731130619561699104832848913402, −7.40062168008255554230750610223, −6.12631350053642314667733471854, −4.22801693916659988908357149931, −0.62006106845296931275641152534, 1.29736469663628919431495227487, 4.36513143069970768194242146457, 6.10162153300897952267190393355, 7.71580167114712307662832463804, 9.426498870051907309052232242429, 10.40127813181405590007222844943, 11.41508613136889967547295600909, 12.31155936753141178152212814376, 14.11926865940305140236198029978, 15.30114460468291199154941329848

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