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Properties

Label 1-456-456.11-r0-0-0
Degree $1$
Conductor $456$
Sign $-0.671 + 0.740i$
Analytic cond. $2.11765$
Root an. cond. $2.11765$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s − 7-s − 11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)35-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s − 7-s − 11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)35-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(2.11765\)
Root analytic conductor: \(2.11765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 456,\ (0:\ ),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07332075036 + 0.1654404953i\)
\(L(\frac12)\) \(\approx\) \(0.07332075036 + 0.1654404953i\)
\(L(1)\) \(\approx\) \(0.6507224917 - 0.04733421595i\)
\(L(1)\) \(\approx\) \(0.6507224917 - 0.04733421595i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.41691720317559709180210806665, −22.85660312238485004405092397378, −22.11143880630326234470935967455, −21.0757802327665878346642441286, −20.21541598663925887491237833532, −19.039744137499816151163825461369, −18.7465801707558517423129284079, −17.83876427701859817370777431763, −16.31470896374468675438115510495, −16.06225393446153697837585780242, −14.98771373540978537277281834178, −14.04244902516157779621415710346, −13.19277884482164807516680627560, −12.160389526228917120744150800636, −11.25707208737772787610267527579, −10.33771288236815435051901394839, −9.54559348179009784390860949146, −8.32906157037330413110722433478, −7.27793667733632542203256898056, −6.57246842303350528208245962089, −5.49946528818964664003465252158, −4.08269865218714847211406586513, −3.19536718765857835775317806675, −2.223982983164208528197671013961, −0.09951084735344480981953384294, 1.46580452940233610687384142422, 3.1081478418318391438779150776, 3.861682368054588955613614953799, 5.256242502697095671717113563100, 5.91432257647601098514882495739, 7.35230626150483701489165334133, 8.13816426523252028836593948765, 9.076711726496954754697965016832, 10.09625676469084504356369402800, 10.949173300639466963081490528226, 12.24913142617713533919528597672, 12.86906428769432710498863859892, 13.4683317160924030813670074480, 14.93892852632145437664245191597, 15.80779461706919686359766548372, 16.28221054273437420234683919396, 17.2926361890363340725437280919, 18.32821072578386129619768453758, 19.24922042196161864811624049207, 20.00954391877229052633841665578, 20.72354664734888851042375886621, 21.67967347659270054076008144815, 22.68576478822599200480761780928, 23.54886751110428469794985071629, 23.99886563727266491679647736302

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