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Properties

Label 1-504-504.187-r0-0-0
Degree $1$
Conductor $504$
Sign $0.975 + 0.220i$
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  + 5-s + 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + 5-s + 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ 0.975 + 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705467629 + 0.1903605810i\)
\(L(\frac12)\) \(\approx\) \(1.705467629 + 0.1903605810i\)
\(L(1)\) \(\approx\) \(1.314848881 + 0.06304660571i\)
\(L(1)\) \(\approx\) \(1.314848881 + 0.06304660571i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
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 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
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

−23.61491569480099947990763562987, −22.6039759424242281415884176122, −21.82621755328228602332037507818, −21.33330979350416265090086154499, −20.027292445778090555625397005040, −19.6464165572141068620868784662, −18.30358805912409651456443148789, −17.61217854243862545302067511016, −16.9873293248992349032376655996, −15.9880115927436116696572120105, −14.81064358732828833757386138812, −14.23403222986703921703563298215, −13.23584346279372429173479311490, −12.45315109809474981656236822516, −11.433986492391984697025521079374, −10.28772470276676758681994520133, −9.67815162821262465423254413197, −8.715105871307706953813662809027, −7.62428122733026323997456276428, −6.466870560126644635963282025643, −5.747261804206828838649970088609, −4.69126817525044601104573656545, −3.41803789353178875313193384381, −2.2848384245399547811006381459, −1.13688126697383077927639925154, 1.31181600399821342518218751340, 2.27558029285428899665603408804, 3.569309711924958997958899055886, 4.74046063146494021790598533328, 5.78049784321640526665880017580, 6.61847452657026656614585068453, 7.59508726398205579384468151453, 8.94784620383816079301220785399, 9.5869668635572266360690600500, 10.35665096764590260686244492264, 11.69230075799299086575366817881, 12.244731776199143996361801772230, 13.54357279283713222749474432997, 14.17810156760251536090378681576, 14.79883912559839633558009116664, 16.35979606748988792264600687702, 16.68992101826537722962690306098, 17.779434257759131054749476673574, 18.46529947999974804761263507599, 19.42863667476662507479694968465, 20.41838821514747700668499027652, 21.12505856635402429123891718073, 22.15045997613591008108860604119, 22.47167385734139687878518018034, 23.84221238741622727723025060222

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