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Properties

Label 2-45-9.7-c3-0-8
Degree $2$
Conductor $45$
Sign $0.185 + 0.982i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.28 − 3.96i)2-s + (3.36 + 3.96i)3-s + (−6.45 − 11.1i)4-s + (−2.5 − 4.33i)5-s + (23.3 − 4.26i)6-s + (10.0 − 17.4i)7-s − 22.4·8-s + (−4.37 + 26.6i)9-s − 22.8·10-s + (−33.1 + 57.4i)11-s + (22.5 − 63.2i)12-s + (23.4 + 40.5i)13-s + (−45.9 − 79.6i)14-s + (8.74 − 24.4i)15-s + (0.237 − 0.411i)16-s − 47.6·17-s + ⋯
L(s)  = 1  + (0.808 − 1.40i)2-s + (0.647 + 0.762i)3-s + (−0.807 − 1.39i)4-s + (−0.223 − 0.387i)5-s + (1.59 − 0.290i)6-s + (0.543 − 0.940i)7-s − 0.993·8-s + (−0.162 + 0.986i)9-s − 0.723·10-s + (−0.909 + 1.57i)11-s + (0.543 − 1.52i)12-s + (0.499 + 0.864i)13-s + (−0.878 − 1.52i)14-s + (0.150 − 0.421i)15-s + (0.00371 − 0.00643i)16-s − 0.679·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.185 + 0.982i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.185 + 0.982i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68109 - 1.39389i\)
\(L(\frac12)\) \(\approx\) \(1.68109 - 1.39389i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.36 - 3.96i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
good2 \( 1 + (-2.28 + 3.96i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-10.0 + 17.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (33.1 - 57.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-23.4 - 40.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 47.6T + 4.91e3T^{2} \)
19 \( 1 + 9.95T + 6.85e3T^{2} \)
23 \( 1 + (4.79 + 8.30i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-89.3 + 154. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (77.0 + 133. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 248.T + 5.06e4T^{2} \)
41 \( 1 + (124. + 216. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-106. + 183. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (237. - 411. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 546.T + 1.48e5T^{2} \)
59 \( 1 + (-209. - 363. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-272. + 472. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-223. - 387. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 409.T + 3.57e5T^{2} \)
73 \( 1 + 358.T + 3.89e5T^{2} \)
79 \( 1 + (-325. + 564. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-406. + 704. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 200.T + 7.04e5T^{2} \)
97 \( 1 + (126. - 218. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.72346723676736811848205319575, −13.71346488984903836479164585017, −12.86535845187315297742508560129, −11.42224610078832679524964013602, −10.48809490749553435021206103738, −9.488151085033153498005509863001, −7.73606568722962367609444067721, −4.72541128899661332326439867098, −4.12354896013801141443488378941, −2.12899469576036895648413423131, 3.15548276009025690581106693405, 5.42319146679755611014026450036, 6.52588188070905658732262697447, 8.046974351732625058273493503609, 8.526792484966013922159737843448, 11.12846891646638637698627006230, 12.72904883014196372868153141460, 13.55759441069901328376699476252, 14.54590654260185934608641950117, 15.36862004878217681134199260617

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