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Properties

Label 2-45-15.8-c3-0-3
Degree $2$
Conductor $45$
Sign $0.915 + 0.402i$
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  + (0.287 + 0.287i)2-s − 7.83i·4-s + (9.93 + 5.12i)5-s + (15.0 − 15.0i)7-s + (4.55 − 4.55i)8-s + (1.38 + 4.33i)10-s + 27.9i·11-s + (2.49 + 2.49i)13-s + 8.66·14-s − 60.0·16-s + (−67.9 − 67.9i)17-s + 95.2i·19-s + (40.1 − 77.8i)20-s + (−8.06 + 8.06i)22-s + (−121. + 121. i)23-s + ⋯
L(s)  = 1  + (0.101 + 0.101i)2-s − 0.979i·4-s + (0.888 + 0.457i)5-s + (0.812 − 0.812i)7-s + (0.201 − 0.201i)8-s + (0.0438 + 0.137i)10-s + 0.767i·11-s + (0.0533 + 0.0533i)13-s + 0.165·14-s − 0.938·16-s + (−0.968 − 0.968i)17-s + 1.14i·19-s + (0.448 − 0.870i)20-s + (−0.0781 + 0.0781i)22-s + (−1.10 + 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.915 + 0.402i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56826 - 0.329203i\)
\(L(\frac12)\) \(\approx\) \(1.56826 - 0.329203i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.93 - 5.12i)T \)
good2 \( 1 + (-0.287 - 0.287i)T + 8iT^{2} \)
7 \( 1 + (-15.0 + 15.0i)T - 343iT^{2} \)
11 \( 1 - 27.9iT - 1.33e3T^{2} \)
13 \( 1 + (-2.49 - 2.49i)T + 2.19e3iT^{2} \)
17 \( 1 + (67.9 + 67.9i)T + 4.91e3iT^{2} \)
19 \( 1 - 95.2iT - 6.85e3T^{2} \)
23 \( 1 + (121. - 121. i)T - 1.21e4iT^{2} \)
29 \( 1 - 99.0T + 2.43e4T^{2} \)
31 \( 1 + 28.7T + 2.97e4T^{2} \)
37 \( 1 + (-271. + 271. i)T - 5.06e4iT^{2} \)
41 \( 1 - 453. iT - 6.89e4T^{2} \)
43 \( 1 + (-30.5 - 30.5i)T + 7.95e4iT^{2} \)
47 \( 1 + (254. + 254. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-224. + 224. i)T - 1.48e5iT^{2} \)
59 \( 1 + 483.T + 2.05e5T^{2} \)
61 \( 1 + 264.T + 2.26e5T^{2} \)
67 \( 1 + (498. - 498. i)T - 3.00e5iT^{2} \)
71 \( 1 + 609. iT - 3.57e5T^{2} \)
73 \( 1 + (74.6 + 74.6i)T + 3.89e5iT^{2} \)
79 \( 1 + 406. iT - 4.93e5T^{2} \)
83 \( 1 + (-652. + 652. i)T - 5.71e5iT^{2} \)
89 \( 1 - 139.T + 7.04e5T^{2} \)
97 \( 1 + (-557. + 557. i)T - 9.12e5iT^{2} \)
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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

−14.94812651851917826639796113769, −14.17917313544055233734597979054, −13.38939197720367274725432656229, −11.45317930489957308507046420915, −10.35436745571068458690911320925, −9.508830653249420744450626528231, −7.45573851934445827110447598526, −6.10387300716754415170400506696, −4.64576086352322068500365936112, −1.76357738060767715789595982688, 2.37670298819219455600856056368, 4.62033173964302376126405572259, 6.24547160018862273088215565104, 8.265837237588132546554847072255, 8.945832281430515735944828900717, 10.82639862147357300214752201291, 12.02651327483015845959389564144, 13.06991283217070793090289958689, 14.03035500858535401648275713498, 15.49829869880911239255953745025

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