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Properties

Label 1-2664-2664.1859-r0-0-0
Degree $1$
Conductor $2664$
Sign $0.999 + 0.0357i$
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  + (0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)17-s + (−0.939 − 0.342i)19-s + (−0.5 + 0.866i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)41-s + (−0.5 + 0.866i)43-s + 47-s + (0.766 + 0.642i)49-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)17-s + (−0.939 − 0.342i)19-s + (−0.5 + 0.866i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)41-s + (−0.5 + 0.866i)43-s + 47-s + (0.766 + 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.999 + 0.0357i$
Analytic conductor: \(12.3715\)
Root analytic conductor: \(12.3715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2664,\ (0:\ ),\ 0.999 + 0.0357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.392152279 + 0.04271782808i\)
\(L(\frac12)\) \(\approx\) \(2.392152279 + 0.04271782808i\)
\(L(1)\) \(\approx\) \(1.430523798 + 0.01860573153i\)
\(L(1)\) \(\approx\) \(1.430523798 + 0.01860573153i\)

Euler product

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

−19.15043858515843142941910113442, −18.52331929443623928722902031120, −17.94719144478814647523542703611, −17.179930136113971894044559424917, −16.745136877798786527468923399302, −15.707829145359020088146656352931, −14.84448678839855499398364708001, −14.40493468851688741957341755213, −13.62579539137273890664979319109, −13.15674688754123945821900236295, −12.102269528546328871453932177068, −11.11872472050025675566398009713, −10.76376196754387411384999854742, −10.21190853927831212267899750898, −8.91793461739798059314073616809, −8.588103636713003095214873639733, −7.66937865663068258596240937323, −6.68792825477700395061764894919, −6.07549198687435835547086683603, −5.48206408328476479425284943073, −4.227970511272712815292074874323, −3.75111617271435468931306307790, −2.54898460251075227767607479091, −1.84443223832000627079631832038, −0.91149429770971107280422207751, 0.99179978507410560622506846202, 1.90689045077371807213009593852, 2.312634375967111346104029964873, 3.813256220965924111692506865077, 4.58581585863807446329918342799, 5.12665430007069753914878323136, 6.11816610248904854050883317279, 6.68618622754781369385212637976, 7.791236333288480496319959103717, 8.52719825008508309327062869133, 9.193206158799322862376308203855, 9.73525887578190566476039367167, 10.79574262963902480517593732848, 11.45361243777480271645742434095, 12.14576667261787547710554587153, 12.97849621741591984068374749852, 13.65003795071146721103751403959, 14.254607303845296201882695572797, 15.09991618171368015478374333538, 15.750911930600973155654752380389, 16.561406038771826109994660132281, 17.5430534225229291354323318062, 17.6447962849930270274758588952, 18.41092245181494536729815546848, 19.40814765454197565275351172165

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