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Properties

Label 1-2664-2664.1757-r0-0-0
Degree $1$
Conductor $2664$
Sign $0.580 - 0.814i$
Analytic cond. $12.3715$
Root an. cond. $12.3715$
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  + (0.342 + 0.939i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (0.866 − 0.5i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.984 − 0.173i)35-s + (0.173 − 0.984i)41-s + (0.866 − 0.5i)43-s − 47-s + (−0.939 − 0.342i)49-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (0.866 − 0.5i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.984 − 0.173i)35-s + (0.173 − 0.984i)41-s + (0.866 − 0.5i)43-s − 47-s + (−0.939 − 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(12.3715\)
Root analytic conductor: \(12.3715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2664,\ (0:\ ),\ 0.580 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.787046923 - 0.9209819060i\)
\(L(\frac12)\) \(\approx\) \(1.787046923 - 0.9209819060i\)
\(L(1)\) \(\approx\) \(1.266748816 - 0.1638616787i\)
\(L(1)\) \(\approx\) \(1.266748816 - 0.1638616787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 - iT \)
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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

−19.52509235190191143822643768867, −18.64350408337510424136261304220, −17.835262058971030677601383112942, −17.53105592396844917309568334416, −16.41068284037348856124638653423, −16.10095365609329026129711336975, −14.99979669114173616437490744887, −14.73099834735813099687370748934, −13.5401869511208129511844953047, −12.95255271862714015977815965166, −12.35928662663850941221966292312, −11.65356751286269128889043840505, −10.89468503145473923506637744420, −9.81835181811317629387615829928, −9.20518341851779447448152958118, −8.70533109146204004548694327002, −7.888757037633846076729405340197, −6.98606431753546536310906392658, −5.89158634056349430563134364341, −5.51270290274360306812294113970, −4.66684344678875691115167897648, −3.77457039121442837459546136306, −2.83514117553307943837729541150, −1.58949991617223102070881323201, −1.33464908362865461855691718704, 0.67173549718373974998255348774, 1.55889409917435191359413257201, 2.71279075005694890055165351182, 3.546481689796046377413064652649, 4.00855360358497443743768083010, 5.339290017495637292301806414004, 5.92843775746629074899377030610, 6.94817190555672716002224360662, 7.25887799296095361383359624348, 8.25051010051955787760978482709, 9.18402345040282907471496268507, 9.851417186507439671800710838170, 10.71902105452544246966028277813, 11.245126587514847676374840599571, 11.727263947105716431958996847135, 13.14091709903948739766057623078, 13.545550300108075140087242480455, 14.309199248583292654644590271118, 14.63237758934733041045029111749, 15.841076919448110921102940799532, 16.366660427100981182624792091617, 17.11756306317553354769745073832, 17.83248355531007965717114770039, 18.65036551958964657639245513354, 18.94928257615967752004089157037

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