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Properties

Label 1-504-504.227-r1-0-0
Degree $1$
Conductor $504$
Sign $0.458 - 0.888i$
Analytic cond. $54.1623$
Root an. cond. $54.1623$
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

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

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(54.1623\)
Root analytic conductor: \(54.1623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (1:\ ),\ 0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.484865966 - 0.9052590184i\)
\(L(\frac12)\) \(\approx\) \(1.484865966 - 0.9052590184i\)
\(L(1)\) \(\approx\) \(1.118223509 - 0.05347224213i\)
\(L(1)\) \(\approx\) \(1.118223509 - 0.05347224213i\)

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

−23.685811787384518723300649200127, −22.61612447957253845022449337035, −21.93444817429284881624109463494, −20.96204383750576813309687835205, −20.134692276217568695320418918989, −19.66036250193912534983666938532, −18.30851010006491840656708338129, −17.45746113236122751958239204895, −16.97825598211016382408747736975, −15.87913213676616013461111497091, −15.02033441392619536148667435324, −14.091465009984393349761765425169, −13.03703719236927671686408591798, −12.4633388717075308145546475399, −11.55142744375069212230881953273, −10.12709577523532161839755120070, −9.69941571445860945913473682927, −8.54267220066059629525878206855, −7.73576727471225734674424107190, −6.50294014281005296891854362636, −5.50877144393834127105187083420, −4.66210652359562385190645229588, −3.547388261364850715626035460442, −2.06646885558468750618648181788, −1.17014627071837424644997580441, 0.45895433648521590135452186879, 2.06850636445850489835157496899, 2.919579309902746109145269597798, 4.12274789740696199393193295016, 5.29768976064756531301271365956, 6.482189104232405977273946139569, 6.95455220915242729903969032079, 8.262250039096383584386671107290, 9.361105809394658846526092286, 10.00780049130328010204927434546, 11.3364912052606592917814971172, 11.582534040551202943574686592175, 13.10971857142138819227977928798, 13.948404139352941858874557788158, 14.46465631344982706525909132240, 15.56706788058748510889874796199, 16.49035970658154305059729241564, 17.37878988446750195273556692752, 18.24419433366967255324241069685, 18.98493406621648898354415312204, 19.77001745916044406757417809938, 20.876616387170625904066852061545, 21.80913142056751029167836323590, 22.2449818435491431721699284057, 23.14367046127090066536869451098

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