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Properties

Label 2-456-24.11-c1-0-23
Degree $2$
Conductor $456$
Sign $0.841 - 0.540i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.450i)2-s + (−1.59 − 0.679i)3-s + (1.59 − 1.20i)4-s + 2.84·5-s + (2.44 + 0.192i)6-s + 0.466i·7-s + (−1.59 + 2.33i)8-s + (2.07 + 2.16i)9-s + (−3.81 + 1.28i)10-s + 4.27i·11-s + (−3.35 + 0.843i)12-s − 4.00i·13-s + (−0.210 − 0.624i)14-s + (−4.53 − 1.93i)15-s + (1.07 − 3.85i)16-s + 3.48i·17-s + ⋯
L(s)  = 1  + (−0.947 + 0.318i)2-s + (−0.919 − 0.392i)3-s + (0.796 − 0.604i)4-s + 1.27·5-s + (0.996 + 0.0784i)6-s + 0.176i·7-s + (−0.562 + 0.826i)8-s + (0.692 + 0.721i)9-s + (−1.20 + 0.405i)10-s + 1.28i·11-s + (−0.969 + 0.243i)12-s − 1.11i·13-s + (−0.0561 − 0.167i)14-s + (−1.17 − 0.499i)15-s + (0.269 − 0.963i)16-s + 0.845i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819814 + 0.240444i\)
\(L(\frac12)\) \(\approx\) \(0.819814 + 0.240444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.450i)T \)
3 \( 1 + (1.59 + 0.679i)T \)
19 \( 1 - T \)
good5 \( 1 - 2.84T + 5T^{2} \)
7 \( 1 - 0.466iT - 7T^{2} \)
11 \( 1 - 4.27iT - 11T^{2} \)
13 \( 1 + 4.00iT - 13T^{2} \)
17 \( 1 - 3.48iT - 17T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 - 8.54T + 29T^{2} \)
31 \( 1 - 7.64iT - 31T^{2} \)
37 \( 1 - 0.360iT - 37T^{2} \)
41 \( 1 - 3.88iT - 41T^{2} \)
43 \( 1 - 6.31T + 43T^{2} \)
47 \( 1 - 3.96T + 47T^{2} \)
53 \( 1 - 1.30T + 53T^{2} \)
59 \( 1 + 3.00iT - 59T^{2} \)
61 \( 1 + 2.20iT - 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 6.80T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−10.67861317160616184014461599539, −10.28880805931311656717510169748, −9.596987996742090065156403285874, −8.397558429220610176217191245869, −7.38943101347114037157158842265, −6.48353497270695118692326167871, −5.77279430027335760106925269667, −4.92312232283425427664648128385, −2.42046289075996728663791411111, −1.31927998120910866263092749604, 0.942908000481427076435144303500, 2.50481223065941951554886471975, 4.07423829608996797751434438609, 5.60102266290291267920352076706, 6.29697224733660098926413227646, 7.18267016932533637987056490581, 8.636727965230262572530147149339, 9.448534092166280824315345145739, 10.01990804632379215220632069091, 10.91116349589657125324960239280

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