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Properties

Label 2-45-45.14-c4-0-4
Degree $2$
Conductor $45$
Sign $-0.952 - 0.305i$
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  + (3.64 + 6.30i)2-s + (−4.56 − 7.75i)3-s + (−18.5 + 32.0i)4-s + (0.540 + 24.9i)5-s + (32.2 − 57.0i)6-s + (−38.4 + 22.2i)7-s − 153.·8-s + (−39.3 + 70.8i)9-s + (−155. + 94.4i)10-s + (142. − 82.2i)11-s + (333. − 2.80i)12-s + (140. + 81.0i)13-s + (−280. − 161. i)14-s + (191. − 118. i)15-s + (−261. − 452. i)16-s − 79.6·17-s + ⋯
L(s)  = 1  + (0.910 + 1.57i)2-s + (−0.507 − 0.861i)3-s + (−1.15 + 2.00i)4-s + (0.0216 + 0.999i)5-s + (0.896 − 1.58i)6-s + (−0.784 + 0.453i)7-s − 2.39·8-s + (−0.485 + 0.874i)9-s + (−1.55 + 0.944i)10-s + (1.17 − 0.679i)11-s + (2.31 − 0.0194i)12-s + (0.830 + 0.479i)13-s + (−1.42 − 0.824i)14-s + (0.850 − 0.525i)15-s + (−1.02 − 1.76i)16-s − 0.275·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ -0.952 - 0.305i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.259562 + 1.65793i\)
\(L(\frac12)\) \(\approx\) \(0.259562 + 1.65793i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.56 + 7.75i)T \)
5 \( 1 + (-0.540 - 24.9i)T \)
good2 \( 1 + (-3.64 - 6.30i)T + (-8 + 13.8i)T^{2} \)
7 \( 1 + (38.4 - 22.2i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-142. + 82.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-140. - 81.0i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 79.6T + 8.35e4T^{2} \)
19 \( 1 - 493.T + 1.30e5T^{2} \)
23 \( 1 + (99.2 - 171. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-91.9 + 53.1i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (517. - 896. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.04e3iT - 1.87e6T^{2} \)
41 \( 1 + (-38.6 - 22.3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.98e3 + 1.72e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (418. + 724. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 307.T + 7.89e6T^{2} \)
59 \( 1 + (3.86e3 + 2.23e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-876. - 1.51e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.52e3 - 2.03e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.45e3iT - 2.54e7T^{2} \)
73 \( 1 - 486. iT - 2.83e7T^{2} \)
79 \( 1 + (-3.63e3 - 6.29e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (1.75e3 + 3.03e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 5.21e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.51e3 - 871. i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.69074356734327636487180106686, −14.16208303486409954580942011879, −13.80342682989707982300706051624, −12.45981424186918970611863817953, −11.34108740268802322214938930479, −8.957248214787583349472635902227, −7.38078027984936327197688588992, −6.47571109733743383423367516342, −5.76651443795118590625742942272, −3.49625242128697346405952500248, 0.938714002075575156350776408470, 3.57593916166412204876148845806, 4.55694335857064574351254386337, 5.94895412527568502726268002060, 9.255392346342109807830126524698, 9.889039592797912591518388812269, 11.20720077949083854368066457351, 12.13466484608866698070715885824, 13.04973620410657583213245088969, 14.18557999195317167225264731244

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