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Properties

Label 2-456-456.11-c1-0-58
Degree $2$
Conductor $456$
Sign $-0.990 - 0.136i$
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  + (−1.07 − 0.923i)2-s + (0.703 − 1.58i)3-s + (0.292 + 1.97i)4-s + (−1.52 + 2.64i)5-s + (−2.21 + 1.04i)6-s − 2.32i·7-s + (1.51 − 2.38i)8-s + (−2.00 − 2.22i)9-s + (4.08 − 1.42i)10-s − 1.59i·11-s + (3.33 + 0.929i)12-s + (−3.06 + 1.77i)13-s + (−2.14 + 2.48i)14-s + (3.11 + 4.28i)15-s + (−3.82 + 1.15i)16-s + (−2.45 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.757 − 0.653i)2-s + (0.406 − 0.913i)3-s + (0.146 + 0.989i)4-s + (−0.683 + 1.18i)5-s + (−0.904 + 0.426i)6-s − 0.878i·7-s + (0.535 − 0.844i)8-s + (−0.669 − 0.742i)9-s + (1.29 − 0.449i)10-s − 0.481i·11-s + (0.963 + 0.268i)12-s + (−0.850 + 0.491i)13-s + (−0.574 + 0.665i)14-s + (0.804 + 1.10i)15-s + (−0.957 + 0.289i)16-s + (−0.596 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.990 - 0.136i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ -0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0314200 + 0.457441i\)
\(L(\frac12)\) \(\approx\) \(0.0314200 + 0.457441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.923i)T \)
3 \( 1 + (-0.703 + 1.58i)T \)
19 \( 1 + (0.478 + 4.33i)T \)
good5 \( 1 + (1.52 - 2.64i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.32iT - 7T^{2} \)
11 \( 1 + 1.59iT - 11T^{2} \)
13 \( 1 + (3.06 - 1.77i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.45 + 1.41i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.62 + 4.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.36 + 5.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.183iT - 31T^{2} \)
37 \( 1 - 9.54iT - 37T^{2} \)
41 \( 1 + (7.72 + 4.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.14 - 3.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.18 + 2.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.76 - 6.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.63 - 4.41i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.6 + 6.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.73 + 4.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.38 + 4.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.471 + 0.816i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.6 + 7.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.26iT - 83T^{2} \)
89 \( 1 + (-13.4 + 7.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.397 + 0.689i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.69710870269163608865744143340, −9.807161168763511683793065133346, −8.680364236011746448133356771072, −7.83295933627510906690706106401, −7.02475878895635156798961313272, −6.64602595944849242400304430284, −4.27026369243747963316733240462, −3.17205743793756707878374978875, −2.25788456373964283894281394888, −0.32679817987797263898228279864, 2.05588856128263999317936938153, 3.92726680979300178960382346135, 5.09896313849774800519752181654, 5.59584992906214834721183763803, 7.29062769523735035497338244435, 8.260872335307009758369525419151, 8.711277334367607386119653386508, 9.558416424922693833383440526385, 10.24607148068273144635423260585, 11.41335257698853089284448313380

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