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Properties

Label 2-45-9.5-c4-0-11
Degree $2$
Conductor $45$
Sign $0.704 + 0.710i$
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  + (6.87 − 3.97i)2-s + (−1.96 + 8.78i)3-s + (23.5 − 40.7i)4-s + (9.68 + 5.59i)5-s + (21.3 + 68.2i)6-s + (10.1 + 17.5i)7-s − 246. i·8-s + (−73.2 − 34.5i)9-s + 88.7·10-s + (−42.4 + 24.4i)11-s + (311. + 286. i)12-s + (−87.2 + 151. i)13-s + (139. + 80.3i)14-s + (−68.1 + 74.0i)15-s + (−602. − 1.04e3i)16-s + 237. i·17-s + ⋯
L(s)  = 1  + (1.71 − 0.992i)2-s + (−0.218 + 0.975i)3-s + (1.47 − 2.54i)4-s + (0.387 + 0.223i)5-s + (0.592 + 1.89i)6-s + (0.206 + 0.357i)7-s − 3.85i·8-s + (−0.904 − 0.426i)9-s + 0.887·10-s + (−0.350 + 0.202i)11-s + (2.16 + 1.99i)12-s + (−0.516 + 0.894i)13-s + (0.710 + 0.410i)14-s + (−0.302 + 0.329i)15-s + (−2.35 − 4.07i)16-s + 0.821i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.710i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.704 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.704 + 0.710i$
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: \((2,\ 45,\ (\ :2),\ 0.704 + 0.710i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.19863 - 1.33299i\)
\(L(\frac12)\) \(\approx\) \(3.19863 - 1.33299i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.96 - 8.78i)T \)
5 \( 1 + (-9.68 - 5.59i)T \)
good2 \( 1 + (-6.87 + 3.97i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (-10.1 - 17.5i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (42.4 - 24.4i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (87.2 - 151. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 237. iT - 8.35e4T^{2} \)
19 \( 1 + 177.T + 1.30e5T^{2} \)
23 \( 1 + (-57.7 - 33.3i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (110. - 63.6i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-326. + 565. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.55e3T + 1.87e6T^{2} \)
41 \( 1 + (-501. - 289. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.11e3 + 1.93e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-534. + 308. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.08e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.54e3 + 890. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (536. + 929. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.95e3 + 6.85e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 7.44e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.92e3T + 2.83e7T^{2} \)
79 \( 1 + (430. + 745. i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-1.26e3 + 733. i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 1.36e4iT - 6.27e7T^{2} \)
97 \( 1 + (5.26e3 + 9.12e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−14.80433598091519308598921250340, −13.82120011206165086227478903668, −12.52012396461707495309293474102, −11.47477265973937341289984262251, −10.54083768457919662825156753308, −9.527536137189095688163586254538, −6.36574336822353682476664338821, −5.18654414694050659311682320482, −4.01236597464988512083885674033, −2.34684294228603413640027223150, 2.73324041109692110084754465668, 4.89157927778747365124218548067, 5.99464275876192468126999087887, 7.22917361468315658096655445110, 8.198957521986087612963096202529, 11.13116456009927156148414158067, 12.34471580201891448113980874156, 13.12201705824749219332396088638, 13.92144868556043351693555032983, 14.89224736284325819758199950682

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