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Properties

Label 1-2736-2736.797-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.999 - 0.0436i$
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  + (0.866 − 0.5i)5-s + (0.5 − 0.866i)7-s + (−0.866 − 0.5i)11-s + (−0.866 + 0.5i)13-s − 17-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.866 − 0.5i)29-s + (0.5 + 0.866i)31-s − i·35-s − i·37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (0.5 − 0.866i)7-s + (−0.866 − 0.5i)11-s + (−0.866 + 0.5i)13-s − 17-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.866 − 0.5i)29-s + (0.5 + 0.866i)31-s − i·35-s − i·37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.999 - 0.0436i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.999 - 0.0436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01505443361 - 0.6899349250i\)
\(L(\frac12)\) \(\approx\) \(0.01505443361 - 0.6899349250i\)
\(L(1)\) \(\approx\) \(0.9219457739 - 0.2846325027i\)
\(L(1)\) \(\approx\) \(0.9219457739 - 0.2846325027i\)

Euler product

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

−19.540232118525756421198856262201, −18.748090416857416414700775813193, −18.08736461735286912065005975810, −17.61141764548641276878903859118, −17.10228497337112854045217190381, −15.82499589088401652037335847921, −15.35155726508991244375685920716, −14.72107093752881746177008139090, −13.98458935962668624455119827737, −13.19223840803210477555242151515, −12.55938354479473079636075360203, −11.79571235217042839901818457822, −10.80839132135628897511173475428, −10.4384195700470422480577266156, −9.26972977281083667295325426332, −9.15408327644743851400666386642, −7.68923297258941909483194237757, −7.49084536058143556637638959276, −6.25398046814962845655624894877, −5.61069533254865402383840507677, −5.05190112069500562970713455914, −4.07883206003639982401970221430, −2.70458137181384103166877873617, −2.41112782421717640513343713264, −1.581997425321215962687261133752, 0.190111702888071482526243482276, 1.37537425450091052147833775711, 2.19705191862436797871008077119, 2.97904858431506508582077654466, 4.41908662996601035776753819715, 4.63385409772313155163944592227, 5.62858969558485350424732469471, 6.423734847148171314353249935646, 7.23046336407095277694240309046, 8.080988934981525350874231802554, 8.73934832470899467012988510365, 9.61794946422243113116907789064, 10.29589101323965190905141228447, 10.88562016542465206051622696768, 11.75553955511027030278157920174, 12.64262682278322696330139678671, 13.380873584140588547059405444938, 13.80647262001238125448995828930, 14.53908664894882734945613051471, 15.34687219548285030214357347201, 16.50792313601851600051291020047, 16.59161549833273611813043230171, 17.67936898781916105516480291329, 17.90135221697043298972995009774, 18.91048703909471563170901506932

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