Properties

Label 1-504-504.5-r0-0-0
Degree $1$
Conductor $504$
Sign $-0.458 - 0.888i$
Analytic cond. $2.34056$
Root an. cond. $2.34056$
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 + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-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)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-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)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.458 - 0.888i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ -0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5468665606 - 0.8970068537i\)
\(L(\frac12)\) \(\approx\) \(0.5468665606 - 0.8970068537i\)
\(L(1)\) \(\approx\) \(0.9175142161 - 0.3547929909i\)
\(L(1)\) \(\approx\) \(0.9175142161 - 0.3547929909i\)

Euler product

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

−23.75392114650284569629535125938, −23.01105773891876364794066525927, −22.262160813326000368839863779163, −21.38795117710788161077746672561, −20.693442751071452831046550869105, −19.58555755522934227289283682166, −18.73669236682980302288958392241, −18.01926162917867035720361979655, −17.23753673902227692982592870653, −16.251072550407766069094770020354, −15.15253449951928032273991300047, −14.549073353560806061625870885212, −13.63050236138058837145563622819, −12.75521026765990431190099988000, −11.62967039965465654831243736896, −10.85162574749981325529684583638, −9.78423677232356954636269824651, −9.27780769134159749920080828585, −7.7020818926042437149245371186, −7.07469651362378263257661945857, −6.078337126203615983383939825757, −5.00225385204544246847725257457, −3.88483577371129064084287308403, −2.579472560897660910585260801493, −1.83397141197649945471788530650, 0.53278654223515923904233391458, 1.96630860794996828811554477875, 3.08913416147578286506810671155, 4.45191973755098712081468981838, 5.34975136747483245156361283388, 6.155743196997879579470672961224, 7.4269958043488734202747123017, 8.56723175100130470207641067453, 9.0312867599261567226907251522, 10.37058950882203635405007939740, 10.94303138291120260321154224697, 12.35995867402027400405641135792, 12.96185366312020272536905864905, 13.6789854302409668413553455409, 14.85511188289402832320967412266, 15.672940881192499428565730790690, 16.71296739901352693464985936181, 17.25160841468394614354747375255, 18.20518010914182143303758676478, 19.18039908179619713236432440767, 20.13931605757503962473332362647, 20.730349035911772023504668852165, 21.82693618554948173480200367054, 22.187734057362916442000084896383, 23.81016081245432094185011866668

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