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Properties

Label 2-45-45.4-c3-0-12
Degree $2$
Conductor $45$
Sign $-0.809 + 0.586i$
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  + (−2.13 + 1.23i)2-s + (1.50 − 4.97i)3-s + (−0.964 + 1.67i)4-s + (−9.64 − 5.65i)5-s + (2.91 + 12.4i)6-s + (−16.6 + 9.62i)7-s − 24.4i·8-s + (−22.4 − 14.9i)9-s + (27.5 + 0.190i)10-s + (−19.9 − 34.5i)11-s + (6.85 + 7.31i)12-s + (0.873 + 0.504i)13-s + (23.7 − 41.0i)14-s + (−42.6 + 39.4i)15-s + (22.4 + 38.8i)16-s + 52.6i·17-s + ⋯
L(s)  = 1  + (−0.754 + 0.435i)2-s + (0.289 − 0.957i)3-s + (−0.120 + 0.208i)4-s + (−0.862 − 0.505i)5-s + (0.198 + 0.848i)6-s + (−0.899 + 0.519i)7-s − 1.08i·8-s + (−0.832 − 0.554i)9-s + (0.871 + 0.00602i)10-s + (−0.546 − 0.946i)11-s + (0.164 + 0.175i)12-s + (0.0186 + 0.0107i)13-s + (0.452 − 0.783i)14-s + (−0.734 + 0.678i)15-s + (0.350 + 0.606i)16-s + 0.750i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.809 + 0.586i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.809 + 0.586i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0912783 - 0.281612i\)
\(L(\frac12)\) \(\approx\) \(0.0912783 - 0.281612i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.50 + 4.97i)T \)
5 \( 1 + (9.64 + 5.65i)T \)
good2 \( 1 + (2.13 - 1.23i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (16.6 - 9.62i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (19.9 + 34.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-0.873 - 0.504i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 52.6iT - 4.91e3T^{2} \)
19 \( 1 - 49.5T + 6.85e3T^{2} \)
23 \( 1 + (-23.7 - 13.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (127. + 220. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-84.3 + 146. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 419. iT - 5.06e4T^{2} \)
41 \( 1 + (199. - 345. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (310. - 179. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-122. + 70.6i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 290. iT - 1.48e5T^{2} \)
59 \( 1 + (-14.3 + 24.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (366. + 634. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-152. - 88.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 802.T + 3.57e5T^{2} \)
73 \( 1 + 512. iT - 3.89e5T^{2} \)
79 \( 1 + (-306. - 530. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (69.9 - 40.4i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 24.0T + 7.04e5T^{2} \)
97 \( 1 + (-1.18e3 + 683. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.22838908995700033621804176469, −13.35058163420101466007256877426, −12.73281379745119593566925085709, −11.55068117879151331190549643793, −9.452627022649917625143884869210, −8.394976187067162745721088934183, −7.62624108414393492144185698141, −6.12917178586702477962663533127, −3.41302617088186401162198137273, −0.28208428292930798175024566095, 3.13625135944909257440436971516, 4.91072703482415414607850429812, 7.26193225527522707453242873327, 8.770411155791736090115705964984, 9.991521933894920952188380423952, 10.56632214575100304835132781457, 11.86151582146366889422675417031, 13.70080098978657411650561547973, 14.88426660917892703981432143105, 15.76822360276163209289711320704

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