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Properties

Label 2-456-19.12-c2-0-10
Degree $2$
Conductor $456$
Sign $0.168 + 0.985i$
Analytic cond. $12.4251$
Root an. cond. $3.52492$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (0.328 + 0.568i)5-s − 2.26·7-s + (1.5 − 2.59i)9-s + 1.81·11-s + (−9.14 − 5.27i)13-s + (−0.985 − 0.568i)15-s + (−9.38 − 16.2i)17-s + (13.5 + 13.3i)19-s + (3.39 − 1.96i)21-s + (0.434 − 0.751i)23-s + (12.2 − 21.2i)25-s + 5.19i·27-s + (−5.65 − 3.26i)29-s − 25.1i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.0656 + 0.113i)5-s − 0.323·7-s + (0.166 − 0.288i)9-s + 0.164·11-s + (−0.703 − 0.406i)13-s + (−0.0656 − 0.0379i)15-s + (−0.552 − 0.956i)17-s + (0.711 + 0.703i)19-s + (0.161 − 0.0934i)21-s + (0.0188 − 0.0326i)23-s + (0.491 − 0.851i)25-s + 0.192i·27-s + (−0.194 − 0.112i)29-s − 0.811i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(12.4251\)
Root analytic conductor: \(3.52492\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1),\ 0.168 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9446439437\)
\(L(\frac12)\) \(\approx\) \(0.9446439437\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 - 0.866i)T \)
19 \( 1 + (-13.5 - 13.3i)T \)
good5 \( 1 + (-0.328 - 0.568i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 2.26T + 49T^{2} \)
11 \( 1 - 1.81T + 121T^{2} \)
13 \( 1 + (9.14 + 5.27i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (9.38 + 16.2i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-0.434 + 0.751i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (5.65 + 3.26i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 25.1iT - 961T^{2} \)
37 \( 1 + 60.4iT - 1.36e3T^{2} \)
41 \( 1 + (-49.3 + 28.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (20.0 + 34.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (7.05 - 12.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-9.28 - 5.36i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-54.3 + 31.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36.5 + 63.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-26.6 - 15.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (53.9 - 31.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (59.4 + 102. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-6.11 + 3.53i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 9.32T + 6.88e3T^{2} \)
89 \( 1 + (-25.1 - 14.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (14.3 - 8.28i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.63254268095376434031017184654, −9.796766530183108901276663933383, −9.105737719098484981815474361680, −7.78593891038090005811687604832, −6.91532816936564340060951265094, −5.86971933005896994642172628359, −4.95562036766438175491422456862, −3.78788294292941138897319106989, −2.43574548348982437822521496783, −0.43826175446019654460352704605, 1.37880371507275081290103058242, 2.93179193158130289031718212490, 4.40667547485630480101640799007, 5.37099434668077970966926414847, 6.51090873168636518809892048326, 7.17114768768820345608866176143, 8.347413974551987349240273172226, 9.340960428448040123087584446895, 10.16365053337933255679680376788, 11.20718380063762736005362237861

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