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Properties

Label 1-2736-2736.2075-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.816 + 0.577i$
Analytic cond. $12.7059$
Root an. cond. $12.7059$
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 + 7-s + (0.866 − 0.5i)11-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)17-s + (−0.766 + 0.642i)23-s + (−0.766 + 0.642i)25-s + (−0.984 + 0.173i)29-s + (0.5 − 0.866i)31-s + (0.342 + 0.939i)35-s + i·37-s + (0.766 + 0.642i)41-s + (0.642 − 0.766i)43-s + (0.173 + 0.984i)47-s + 49-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)5-s + 7-s + (0.866 − 0.5i)11-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)17-s + (−0.766 + 0.642i)23-s + (−0.766 + 0.642i)25-s + (−0.984 + 0.173i)29-s + (0.5 − 0.866i)31-s + (0.342 + 0.939i)35-s + i·37-s + (0.766 + 0.642i)41-s + (0.642 − 0.766i)43-s + (0.173 + 0.984i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.190498511 + 0.6962771831i\)
\(L(\frac12)\) \(\approx\) \(2.190498511 + 0.6962771831i\)
\(L(1)\) \(\approx\) \(1.366458367 + 0.2082151045i\)
\(L(1)\) \(\approx\) \(1.366458367 + 0.2082151045i\)

Euler product

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

−19.35174808314673765554275103506, −18.22509604519418442764691761900, −17.75818679467607108236810955877, −17.16422040552491942554244091777, −16.43902033189805126622059188275, −15.732520185664462625361807805, −14.89606402153775770693656556148, −14.17745248195330598030647319074, −13.58691749145796033437463915679, −12.706530679132993234337288591592, −12.14292131178036772606686540659, −11.31529513565310575433583697387, −10.68887006876036665800057686029, −9.72550664755662944648341229713, −8.8543469346113879148194538472, −8.55206688777868809314790662920, −7.67328798496554701388233788185, −6.66699878448741772185941147030, −5.90350446841311595036475677174, −5.16235840967323645928577952480, −4.14017009612194481146131390993, −3.98837253359722747058635685794, −2.23993584841924388401801846036, −1.71916190048008826641831454788, −0.86146861449118466077158305324, 1.02450742042097193297485518723, 1.89938027483732230858587761917, 2.731066469797264851717138826632, 3.73821349703215744312864241971, 4.3267469663913506540342937482, 5.55584361686372359561310026526, 6.05957814388408065112880385569, 6.90422945764629164843205231037, 7.660482733280151249433801997077, 8.459702661410835447939992116684, 9.22369967327337997896246093749, 10.00007022472843741457728708839, 11.01544295231530287793231814157, 11.28015206149472958674580156660, 11.90596293350968420007725271427, 13.198490365272150308862441890625, 13.80029562882159922216084171887, 14.2593381338636947155464255176, 15.09363291593626753029964334467, 15.61950330589182026379440070939, 16.61128177898353968487328341226, 17.370416459637941248527615718498, 18.00350676795820993250727137555, 18.46954711228759824932479273524, 19.2243654523937869786450702725

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