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Properties

Label 1-2664-2664.1939-r0-0-0
Degree $1$
Conductor $2664$
Sign $0.0403 - 0.999i$
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  + (−0.984 + 0.173i)5-s + (−0.766 + 0.642i)7-s + (0.5 + 0.866i)11-s + (0.642 + 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.866 − 0.5i)23-s + (0.939 − 0.342i)25-s + (−0.866 + 0.5i)29-s + (−0.866 + 0.5i)31-s + (0.642 − 0.766i)35-s + (−0.766 + 0.642i)41-s + (0.866 + 0.5i)43-s − 47-s + (0.173 − 0.984i)49-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)5-s + (−0.766 + 0.642i)7-s + (0.5 + 0.866i)11-s + (0.642 + 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.866 − 0.5i)23-s + (0.939 − 0.342i)25-s + (−0.866 + 0.5i)29-s + (−0.866 + 0.5i)31-s + (0.642 − 0.766i)35-s + (−0.766 + 0.642i)41-s + (0.866 + 0.5i)43-s − 47-s + (0.173 − 0.984i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.0403 - 0.999i$
Analytic conductor: \(12.3715\)
Root analytic conductor: \(12.3715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2664,\ (0:\ ),\ 0.0403 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1906315924 - 0.1830900671i\)
\(L(\frac12)\) \(\approx\) \(0.1906315924 - 0.1830900671i\)
\(L(1)\) \(\approx\) \(0.6652864734 + 0.1192169928i\)
\(L(1)\) \(\approx\) \(0.6652864734 + 0.1192169928i\)

Euler product

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

−19.30196957627077499001190654670, −19.12122215280164119180832183221, −18.13435880214390896334295403760, −17.24950627993961708792357320578, −16.49601401786174754985759019315, −16.078909843337441442088465218315, −15.31680928073547092905885950715, −14.65108006419085389678935752564, −13.56451707733912135922909954921, −13.19662438622820251868532421993, −12.34976257883610418630367086307, −11.566907960632719100597970319086, −10.8666313241651617362182927025, −10.30542114663319044235511646622, −9.22457179113628545408899132944, −8.57300221661232787318111601387, −7.84558202317239355321071441585, −7.12886960872548891176009240035, −6.207645346012846800791747341465, −5.65786333887696161458068040900, −4.30523627078687243288636197953, −3.76586470171044761088375467279, −3.302503271961096570159389601, −1.97401273652966934557481962617, −0.78806052481734039994646992731, 0.11060818215294227459983451383, 1.68744832914478069425175947281, 2.4970779106077098366783429031, 3.50305806785623551361229618730, 4.18061343167409313971879145875, 4.85837900150681045468471188972, 6.09036630103969039935185672972, 6.76503650965376221477159947140, 7.22409125396587814681540139000, 8.42648592452183451150096875710, 8.92324024886450142921329823005, 9.61166245736259204499387522154, 10.62473533734143486713255161077, 11.38053016461055637758759362213, 11.91268552466816847813899642681, 12.732298142473898146763019694383, 13.23065733262065286009399483815, 14.39206045500065669292248698420, 14.96232851631854106235256849726, 15.64565824910526742648966977947, 16.20859729029012916002802416599, 16.84173663828873345195793699339, 18.05045489722883003089868100804, 18.32507834273414531839290920979, 19.33795187642547868217535813690

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