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Properties

Label 1-6004-6004.1319-r0-0-0
Degree $1$
Conductor $6004$
Sign $0.866 + 0.499i$
Analytic cond. $27.8824$
Root an. cond. $27.8824$
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)3-s + 5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $0.866 + 0.499i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ 0.866 + 0.499i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.868498666 + 0.4996746105i\)
\(L(\frac12)\) \(\approx\) \(1.868498666 + 0.4996746105i\)
\(L(1)\) \(\approx\) \(1.136614552 + 0.1587463184i\)
\(L(1)\) \(\approx\) \(1.136614552 + 0.1587463184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
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

−17.76151546138434684801788775616, −17.06632378960571520654346109316, −16.80417047301351578572576673030, −15.62742483655186320163612330226, −14.78376441195438139930978764458, −14.49429232210148385249455041862, −13.680082833032614218395517326, −12.76610876981989865991110757869, −12.576684304214778292482521355345, −11.883931292729960253677228689516, −11.08742580256068895924461819344, −10.48021791629982004578162150846, −9.5727565889696853923939056582, −9.075156210841806597911594423930, −8.215090006302719180684532404977, −7.54014054168303993495977376924, −6.69811131448688768312133641499, −6.23480631979541887941791500702, −5.509815811150851280968066311267, −4.9123542234660702960368875280, −4.184460744327272998234524085115, −2.66698311281231812156466043240, −2.152259693925950154552707453508, −1.80916463910182230893295987533, −0.661953442896617340497208634005, 0.80766481368263188241981477903, 1.44498279641089037832109339137, 2.716969373978910425851662903419, 3.24359801429489390700100427232, 4.31042131036218339114279267457, 4.82174979007802475243448649398, 5.4459166631230255172170518264, 6.16459580700712740301804386016, 6.93156523828926870085580322181, 7.54187930460521811467502519246, 8.80075726456739376511099742638, 9.13990047275098714979760241960, 9.85968126891703135672411504660, 10.550535859832374170489878722825, 10.969577910272707509954914059651, 11.69971770062328405769969556068, 12.37804960367066325144282177966, 13.42717262632597125859794191239, 13.8722502648158246060019256760, 14.50652269452103388239886288278, 15.028060818228483588666363228232, 16.03325613042993302698168257822, 16.6019293926292571475986812434, 17.02563669003551187714523486061, 17.701420218847712643554634876503

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