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Properties

Label 2-45-9.5-c4-0-2
Degree $2$
Conductor $45$
Sign $-0.999 + 0.0167i$
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  + (−1.06 + 0.615i)2-s + (1.63 + 8.84i)3-s + (−7.24 + 12.5i)4-s + (−9.68 − 5.59i)5-s + (−7.19 − 8.42i)6-s + (−21.1 − 36.6i)7-s − 37.5i·8-s + (−75.6 + 28.9i)9-s + 13.7·10-s + (23.9 − 13.8i)11-s + (−122. − 43.5i)12-s + (−118. + 204. i)13-s + (45.0 + 26.0i)14-s + (33.6 − 94.8i)15-s + (−92.7 − 160. i)16-s + 304. i·17-s + ⋯
L(s)  = 1  + (−0.266 + 0.153i)2-s + (0.181 + 0.983i)3-s + (−0.452 + 0.784i)4-s + (−0.387 − 0.223i)5-s + (−0.199 − 0.234i)6-s + (−0.431 − 0.746i)7-s − 0.586i·8-s + (−0.933 + 0.357i)9-s + 0.137·10-s + (0.197 − 0.114i)11-s + (−0.853 − 0.302i)12-s + (−0.700 + 1.21i)13-s + (0.229 + 0.132i)14-s + (0.149 − 0.421i)15-s + (−0.362 − 0.627i)16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.999 + 0.0167i$
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: \((2,\ 45,\ (\ :2),\ -0.999 + 0.0167i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00510568 - 0.609507i\)
\(L(\frac12)\) \(\approx\) \(0.00510568 - 0.609507i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 - 8.84i)T \)
5 \( 1 + (9.68 + 5.59i)T \)
good2 \( 1 + (1.06 - 0.615i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (21.1 + 36.6i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-23.9 + 13.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (118. - 204. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 304. iT - 8.35e4T^{2} \)
19 \( 1 + 315.T + 1.30e5T^{2} \)
23 \( 1 + (-728. - 420. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-672. + 388. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (833. - 1.44e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 209.T + 1.87e6T^{2} \)
41 \( 1 + (1.82e3 + 1.05e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-805. - 1.39e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (320. - 185. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.52e3iT - 7.89e6T^{2} \)
59 \( 1 + (-1.19e3 - 688. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (400. + 693. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.19e3 - 2.06e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.97e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.03e4T + 2.83e7T^{2} \)
79 \( 1 + (2.91e3 + 5.05e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (147. - 85.1i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 1.11e4iT - 6.27e7T^{2} \)
97 \( 1 + (-516. - 894. i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.89202226866444645274277320258, −14.61160276581224998758834843902, −13.41875653957273088716747523670, −12.13920132485186655561107097918, −10.70788623722138010859006489996, −9.401494919750860258915867793101, −8.494165067378845220444190986217, −6.99227846650700554486009376467, −4.59641747323991142612759429498, −3.55248023201124252517924527772, 0.40787664666707611138330033269, 2.60874266148221231227526036861, 5.31912984263773671457564642004, 6.77302524657192483987451397123, 8.313621955625590621267854506673, 9.448938942967147189014559367655, 10.92683533701610429844503178075, 12.26289108743115625704602893165, 13.23208902782514400469668983275, 14.60256037658778276911280296236

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