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Properties

Label 2-45-9.4-c3-0-9
Degree $2$
Conductor $45$
Sign $-0.519 + 0.854i$
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  + (−1.52 − 2.63i)2-s + (4.01 − 3.29i)3-s + (−0.626 + 1.08i)4-s + (2.5 − 4.33i)5-s + (−14.7 − 5.57i)6-s + (6.85 + 11.8i)7-s − 20.5·8-s + (5.29 − 26.4i)9-s − 15.2·10-s + (−15.9 − 27.5i)11-s + (1.05 + 6.42i)12-s + (−29.1 + 50.4i)13-s + (20.8 − 36.1i)14-s + (−4.21 − 25.6i)15-s + (36.2 + 62.7i)16-s + 109.·17-s + ⋯
L(s)  = 1  + (−0.537 − 0.931i)2-s + (0.773 − 0.633i)3-s + (−0.0782 + 0.135i)4-s + (0.223 − 0.387i)5-s + (−1.00 − 0.379i)6-s + (0.370 + 0.641i)7-s − 0.907·8-s + (0.196 − 0.980i)9-s − 0.480·10-s + (−0.435 − 0.755i)11-s + (0.0254 + 0.154i)12-s + (−0.621 + 1.07i)13-s + (0.398 − 0.689i)14-s + (−0.0725 − 0.441i)15-s + (0.566 + 0.980i)16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.519 + 0.854i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.519 + 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.651663 - 1.15933i\)
\(L(\frac12)\) \(\approx\) \(0.651663 - 1.15933i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.01 + 3.29i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
good2 \( 1 + (1.52 + 2.63i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-6.85 - 11.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (15.9 + 27.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (29.1 - 50.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 109.T + 4.91e3T^{2} \)
19 \( 1 - 129.T + 6.85e3T^{2} \)
23 \( 1 + (39.8 - 68.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (4.51 + 7.82i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-16.6 + 28.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 22.1T + 5.06e4T^{2} \)
41 \( 1 + (60.8 - 105. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (5.07 + 8.78i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-220. - 382. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 593.T + 1.48e5T^{2} \)
59 \( 1 + (221. - 383. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.2 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (431. - 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 818.T + 3.57e5T^{2} \)
73 \( 1 - 495.T + 3.89e5T^{2} \)
79 \( 1 + (585. + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-212. - 367. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + (799. + 1.38e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.69626722977014724904532432512, −13.75693813878934045149352603924, −12.25200701271730373299165449537, −11.64400251956890117066547162037, −9.840691683484062502654202519277, −9.036613662947798325911057076373, −7.74032378992207525114903483569, −5.74726441312855407022796551994, −2.99063654928726064667389209722, −1.42071591867415135479339662486, 3.12328310932943309067631637485, 5.29571373670886176383503093057, 7.41020051679408249677459222027, 7.934998135984143356986662094374, 9.596284569779329271671793527668, 10.40263303351662260422676745655, 12.30739785273231418545894256313, 13.96941751201069453195496181728, 14.81175009596111253401394284425, 15.69608909398158460441169245540

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