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Properties

Label 1-2808-2808.2245-r0-0-0
Degree $1$
Conductor $2808$
Sign $0.669 + 0.742i$
Analytic cond. $13.0402$
Root an. cond. $13.0402$
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  + (0.939 + 0.342i)5-s + (0.766 + 0.642i)7-s + (−0.766 − 0.642i)11-s + 17-s + (0.5 + 0.866i)19-s + (0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (0.173 + 0.984i)31-s + (0.5 + 0.866i)35-s + (0.5 − 0.866i)37-s + (−0.939 − 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.173 − 0.984i)47-s + (0.173 + 0.984i)49-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)5-s + (0.766 + 0.642i)7-s + (−0.766 − 0.642i)11-s + 17-s + (0.5 + 0.866i)19-s + (0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (0.173 + 0.984i)31-s + (0.5 + 0.866i)35-s + (0.5 − 0.866i)37-s + (−0.939 − 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.173 − 0.984i)47-s + (0.173 + 0.984i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(13.0402\)
Root analytic conductor: \(13.0402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (2245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (0:\ ),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.143629609 + 0.9536425373i\)
\(L(\frac12)\) \(\approx\) \(2.143629609 + 0.9536425373i\)
\(L(1)\) \(\approx\) \(1.387996863 + 0.2441946194i\)
\(L(1)\) \(\approx\) \(1.387996863 + 0.2441946194i\)

Euler product

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

−18.93256660582162455486557314488, −18.37309332939777541888957031418, −17.47326180183363988252741400867, −17.22975412768930089052835150721, −16.506694461927202669113186440522, −15.44430254245274763088702945446, −14.946763378460613528423553731866, −13.9597143653205774523321018097, −13.526482497110635571639681392011, −12.90493856202827147014567808489, −11.986012347006907787796270759670, −11.21086604561371195173162512818, −10.43640866898061968840489136557, −9.75467207764106914982169959565, −9.2322111275694102083794514967, −8.05187922727006810539531587180, −7.64862153287240616196841127524, −6.733315516902136107135131959972, −5.79638462422526559314315139558, −4.99589239186216815435627504126, −4.62572460118296697094274880360, −3.38455393330434447108902219300, −2.456558752434322874806256871360, −1.623816543379931698716580574127, −0.798781741618047505840609868060, 1.13424818616278367746412937117, 1.87951221674037542640710896776, 2.848082731535008381270647616019, 3.39181594394508250752624601709, 4.8475255640482659239841327211, 5.38666528078547261594666595566, 5.9331702058427149503500080545, 6.88221437381329861420863339759, 7.78697012084709023614761689189, 8.47416608699798935705810602383, 9.200251387903365422014482604327, 10.06815784642544078505204100823, 10.69142289823418834626482076213, 11.32066410695603928708669314228, 12.32897828656462779395152723897, 12.82806205591042114329381740249, 13.8542198883072666736590726895, 14.322215178443940530772235521031, 14.87883392137079401206950340922, 15.81780747909204488782834807898, 16.573013477788763145820444824695, 17.18739770472956427460110343241, 18.1847995307338552183907079990, 18.41527087281446962788101607663, 18.98331338473501841597901523631

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