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Properties

Label 2-45-9.2-c4-0-3
Degree $2$
Conductor $45$
Sign $0.399 - 0.916i$
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 + (3.54 + 8.27i)3-s + (−1.00 − 1.74i)4-s + (9.68 − 5.59i)5-s + (3.96 − 33.4i)6-s + (−19.7 + 34.1i)7-s + 67.3i·8-s + (−55.8 + 58.7i)9-s − 41.8·10-s + (168. + 97.4i)11-s + (10.8 − 14.5i)12-s + (48.0 + 83.1i)13-s + (127. − 73.6i)14-s + (80.5 + 60.2i)15-s + (109. − 190. i)16-s + 404. i·17-s + ⋯
L(s)  = 1  + (−0.809 − 0.467i)2-s + (0.394 + 0.918i)3-s + (−0.0630 − 0.109i)4-s + (0.387 − 0.223i)5-s + (0.110 − 0.928i)6-s + (−0.402 + 0.696i)7-s + 1.05i·8-s + (−0.688 + 0.724i)9-s − 0.418·10-s + (1.39 + 0.805i)11-s + (0.0755 − 0.101i)12-s + (0.284 + 0.492i)13-s + (0.651 − 0.375i)14-s + (0.358 + 0.267i)15-s + (0.428 − 0.742i)16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.399 - 0.916i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.830370 + 0.543918i\)
\(L(\frac12)\) \(\approx\) \(0.830370 + 0.543918i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.54 - 8.27i)T \)
5 \( 1 + (-9.68 + 5.59i)T \)
good2 \( 1 + (3.23 + 1.86i)T + (8 + 13.8i)T^{2} \)
7 \( 1 + (19.7 - 34.1i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-168. - 97.4i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-48.0 - 83.1i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 404. iT - 8.35e4T^{2} \)
19 \( 1 + 559.T + 1.30e5T^{2} \)
23 \( 1 + (-675. + 390. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (414. + 239. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (311. + 539. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + (-87.2 + 50.3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-299. + 518. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.85e3 + 1.06e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.45e3iT - 7.89e6T^{2} \)
59 \( 1 + (-3.61e3 + 2.08e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (345. - 598. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.71e3 - 4.71e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 199. iT - 2.54e7T^{2} \)
73 \( 1 + 5.15e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.76e3 + 4.79e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-825. - 476. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 8.55e3iT - 6.27e7T^{2} \)
97 \( 1 + (-2.55e3 + 4.42e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.01827963374055129859228692486, −14.58888828430314653288436191864, −12.87146550587764371477338934367, −11.32983606720422347268380323884, −10.20339689771785730628352523429, −9.178933424268326509741971383063, −8.646320848293268284024304500142, −6.10454879421942938441339007762, −4.32689111603151856147563698223, −2.03879182816610483744365298720, 0.834119373707520900307349180501, 3.43881846966472285703472505636, 6.41777268056955140052020052563, 7.22929607200529489800746386009, 8.630058746451226803283764181387, 9.502110530692893299561287234616, 11.24006487727219648665260816073, 12.83748567597050652140556486750, 13.61511991403300761645077246510, 14.78269338336503131568993378827

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