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Properties

Label 2-45-45.14-c4-0-1
Degree $2$
Conductor $45$
Sign $-0.874 + 0.484i$
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  + (2.61 + 4.53i)2-s + (−8.42 + 3.16i)3-s + (−5.69 + 9.86i)4-s + (−23.1 − 9.45i)5-s + (−36.4 − 29.8i)6-s + (−50.0 + 28.9i)7-s + 24.1·8-s + (60.9 − 53.3i)9-s + (−17.6 − 129. i)10-s + (−146. + 84.5i)11-s + (16.7 − 101. i)12-s + (99.1 + 57.2i)13-s + (−262. − 151. i)14-s + (224. + 6.34i)15-s + (154. + 267. i)16-s − 209.·17-s + ⋯
L(s)  = 1  + (0.654 + 1.13i)2-s + (−0.935 + 0.352i)3-s + (−0.356 + 0.616i)4-s + (−0.925 − 0.378i)5-s + (−1.01 − 0.830i)6-s + (−1.02 + 0.589i)7-s + 0.376·8-s + (0.752 − 0.659i)9-s + (−0.176 − 1.29i)10-s + (−1.21 + 0.699i)11-s + (0.116 − 0.702i)12-s + (0.586 + 0.338i)13-s + (−1.33 − 0.771i)14-s + (0.999 + 0.0281i)15-s + (0.602 + 1.04i)16-s − 0.726·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.874 + 0.484i$
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: \((2,\ 45,\ (\ :2),\ -0.874 + 0.484i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.168779 - 0.652668i\)
\(L(\frac12)\) \(\approx\) \(0.168779 - 0.652668i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.42 - 3.16i)T \)
5 \( 1 + (23.1 + 9.45i)T \)
good2 \( 1 + (-2.61 - 4.53i)T + (-8 + 13.8i)T^{2} \)
7 \( 1 + (50.0 - 28.9i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (146. - 84.5i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-99.1 - 57.2i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 209.T + 8.35e4T^{2} \)
19 \( 1 - 26.9T + 1.30e5T^{2} \)
23 \( 1 + (458. - 794. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-308. + 178. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-629. + 1.08e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 368. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.59e3 - 918. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-277. + 160. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.22e3 + 2.12e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.89e3T + 7.89e6T^{2} \)
59 \( 1 + (4.20e3 + 2.42e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (473. + 820. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.87e3 - 1.08e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 9.07e3iT - 2.54e7T^{2} \)
73 \( 1 - 984. iT - 2.83e7T^{2} \)
79 \( 1 + (-5.12e3 - 8.87e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-2.39e3 - 4.14e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 858. iT - 6.27e7T^{2} \)
97 \( 1 + (3.69e3 - 2.13e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.70668383422228682054527707153, −15.30519848139827905997554863769, −13.35088850047646493262891911731, −12.49452149373280602711689472600, −11.20193143615345987422654079432, −9.688396905445066129466456743711, −7.83895571041815773350127879576, −6.53407331160371875531932144990, −5.38438924148785296572415643003, −4.11179097737919055747855196177, 0.38036039982740127658405237464, 3.03272932490794894914382044195, 4.51911449524402885284007308306, 6.42569486776984230837464082441, 7.87315433185981664760413710255, 10.55096666572470297312866320615, 10.75628633573132894534356807315, 12.12702958645203413797503321727, 12.90585323986205141486288532572, 13.81220062769520301217877896367

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