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Properties

Label 2-45-5.4-c3-0-3
Degree $2$
Conductor $45$
Sign $0.724 + 0.688i$
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  − 1.70i·2-s + 5.10·4-s + (8.10 + 7.70i)5-s − 22.2i·7-s − 22.2i·8-s + (13.1 − 13.7i)10-s + 1.79·11-s + 58.2i·13-s − 37.7·14-s + 2.89·16-s + 18.9i·17-s − 104.·19-s + (41.3 + 39.3i)20-s − 3.04i·22-s + 49.6i·23-s + ⋯
L(s)  = 1  − 0.601i·2-s + 0.638·4-s + (0.724 + 0.688i)5-s − 1.19i·7-s − 0.985i·8-s + (0.414 − 0.436i)10-s + 0.0490·11-s + 1.24i·13-s − 0.721·14-s + 0.0452·16-s + 0.270i·17-s − 1.26·19-s + (0.462 + 0.439i)20-s − 0.0295i·22-s + 0.449i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.724 + 0.688i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.724 + 0.688i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52326 - 0.608324i\)
\(L(\frac12)\) \(\approx\) \(1.52326 - 0.608324i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-8.10 - 7.70i)T \)
good2 \( 1 + 1.70iT - 8T^{2} \)
7 \( 1 + 22.2iT - 343T^{2} \)
11 \( 1 - 1.79T + 1.33e3T^{2} \)
13 \( 1 - 58.2iT - 2.19e3T^{2} \)
17 \( 1 - 18.9iT - 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 - 49.6iT - 1.21e4T^{2} \)
29 \( 1 + 293.T + 2.43e4T^{2} \)
31 \( 1 - 64.4T + 2.97e4T^{2} \)
37 \( 1 - 19.8iT - 5.06e4T^{2} \)
41 \( 1 - 165.T + 6.89e4T^{2} \)
43 \( 1 + 247. iT - 7.95e4T^{2} \)
47 \( 1 - 384. iT - 1.03e5T^{2} \)
53 \( 1 + 463. iT - 1.48e5T^{2} \)
59 \( 1 + 73.7T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 173. iT - 3.00e5T^{2} \)
71 \( 1 - 594.T + 3.57e5T^{2} \)
73 \( 1 - 320. iT - 3.89e5T^{2} \)
79 \( 1 - 770.T + 4.93e5T^{2} \)
83 \( 1 + 173. iT - 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 384. iT - 9.12e5T^{2} \)
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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

−15.04580414434121457709823560752, −13.95321547005588313685926638320, −12.88893923489583463629117815319, −11.31818831405434333774152808174, −10.61409483093948582862124281204, −9.516381846573966986457400854953, −7.31014132394159855816467660097, −6.34860308910963547672125058706, −3.84929438339693226768449064244, −1.92169493751231708746283251778, 2.33154417204343144299479224930, 5.32367951567297272907439055948, 6.21122371265295243962762125332, 8.013020897614252540711249848269, 9.120732485845489073210119384410, 10.69257677770688124698805160245, 12.13926420999235073893479162114, 13.08749260659829955328179777460, 14.71657868965586795843186876137, 15.44199518919444576870156762333

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