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Properties

Label 1-624-624.581-r1-0-0
Degree $1$
Conductor $624$
Sign $-0.926 - 0.376i$
Analytic cond. $67.0581$
Root an. cond. $67.0581$
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  − i·5-s + (0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s − 25-s + (0.866 − 0.5i)29-s + 31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  − i·5-s + (0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s − 25-s + (0.866 − 0.5i)29-s + 31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.926 - 0.376i$
Analytic conductor: \(67.0581\)
Root analytic conductor: \(67.0581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 624,\ (1:\ ),\ -0.926 - 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2759540339 - 1.410939840i\)
\(L(\frac12)\) \(\approx\) \(0.2759540339 - 1.410939840i\)
\(L(1)\) \(\approx\) \(0.9486583728 - 0.4190239904i\)
\(L(1)\) \(\approx\) \(0.9486583728 - 0.4190239904i\)

Euler product

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

−23.28441170367495022605408646524, −21.9815750038777491143111752724, −21.7425884103180143675788243755, −20.87264064267791657549111574659, −19.6569631029670186858944102515, −18.940895911931722966141197131, −18.117057260152376549369541146043, −17.70936497067856823065963220461, −16.34723456845089873229929849281, −15.48267931555432072371569118493, −14.89454945523855314040874311545, −13.97850110454367261541715229966, −13.166839823985555729709274745462, −11.928976335685306788437776463664, −11.359244968132881441418732818357, −10.393520184672681702971420626514, −9.61698668733756187385642636211, −8.29266261903023070893279950344, −7.7919951771040786855044916793, −6.527169554316098728082072211472, −5.73268065450846089895666075536, −4.7985928675369289886045725272, −3.30529361094755871894171002784, −2.68796246214526400526265347210, −1.43942616646002159506090196583, 0.35556356125147757326464795392, 1.29307513213006713205499144534, 2.564822802235910401566036177256, 3.97147374008090017720199791751, 4.81292241970828780956825311753, 5.51085785051338283670381594303, 6.921267467484338798303596133412, 7.86606370241792015971729223950, 8.433466855517333313391207593146, 9.81107864694915972647161119268, 10.228901253249842643157394692487, 11.5738450714545030860146789107, 12.21714610921544392727521188018, 13.26838170482983851693202323828, 13.861802253937247024586219564434, 14.847175270740167863513194130833, 16.084209308115119687435542738749, 16.39567956105647654516203718163, 17.53371909274965039102602017511, 18.06358148802619854078076367793, 19.2037180910321058194922077522, 20.3062408678350484678079450658, 20.60655857140818400898050695707, 21.29503660890568479547097350496, 22.62570163658096619114391530315

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