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Properties

Label 2-45-5.3-c4-0-4
Degree $2$
Conductor $45$
Sign $0.916 - 0.400i$
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  + (1.58 − 1.58i)2-s + 11i·4-s + (20.5 + 14.2i)5-s + (−5 + 5i)7-s + (42.6 + 42.6i)8-s + (55 − 10i)10-s + 173.·11-s + (−110 − 110i)13-s + 15.8i·14-s − 40.9·16-s + (−3.16 + 3.16i)17-s + 198i·19-s + (−156. + 226. i)20-s + (275 − 275i)22-s + (−534. − 534. i)23-s + ⋯
L(s)  = 1  + (0.395 − 0.395i)2-s + 0.687i·4-s + (0.822 + 0.569i)5-s + (−0.102 + 0.102i)7-s + (0.667 + 0.667i)8-s + (0.550 − 0.100i)10-s + 1.43·11-s + (−0.650 − 0.650i)13-s + 0.0806i·14-s − 0.160·16-s + (−0.0109 + 0.0109i)17-s + 0.548i·19-s + (−0.391 + 0.565i)20-s + (0.568 − 0.568i)22-s + (−1.01 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.916 - 0.400i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.01725 + 0.421194i\)
\(L(\frac12)\) \(\approx\) \(2.01725 + 0.421194i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-20.5 - 14.2i)T \)
good2 \( 1 + (-1.58 + 1.58i)T - 16iT^{2} \)
7 \( 1 + (5 - 5i)T - 2.40e3iT^{2} \)
11 \( 1 - 173.T + 1.46e4T^{2} \)
13 \( 1 + (110 + 110i)T + 2.85e4iT^{2} \)
17 \( 1 + (3.16 - 3.16i)T - 8.35e4iT^{2} \)
19 \( 1 - 198iT - 1.30e5T^{2} \)
23 \( 1 + (534. + 534. i)T + 2.79e5iT^{2} \)
29 \( 1 + 996. iT - 7.07e5T^{2} \)
31 \( 1 + 1.19e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.81e3 + 1.81e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.54e3T + 2.82e6T^{2} \)
43 \( 1 + (1.31e3 + 1.31e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-2.23e3 + 2.23e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-1.61e3 - 1.61e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 996. iT - 1.21e7T^{2} \)
61 \( 1 + 1.37e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.87e3 + 1.87e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.89e3T + 2.54e7T^{2} \)
73 \( 1 + (3.90e3 + 3.90e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 612iT - 3.89e7T^{2} \)
83 \( 1 + (4.76e3 + 4.76e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.13e4iT - 6.27e7T^{2} \)
97 \( 1 + (7.20e3 - 7.20e3i)T - 8.85e7iT^{2} \)
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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

−14.76109091192438751759100357981, −13.97956112553481561813897427953, −12.75404722855667310466462215625, −11.80193952771955852279494364350, −10.47840878618667146725695396326, −9.144430301314025351405215853671, −7.50097736048996578703036362525, −5.98293104291306731332285174047, −3.98522224085245742366018525960, −2.32610087388951227424866865486, 1.49249365815959785395890319626, 4.42763178341241295579878515372, 5.81782662249336006436176634683, 6.96106612645856761042494444078, 9.132610635312828922147333953141, 9.892653905701062447475481478538, 11.52735452803774627710944947804, 12.96963973054951225966163362145, 14.05830297737558384419233613909, 14.71115301922596355304987990097

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