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Properties

Label 1-2736-2736.67-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.692 + 0.721i$
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  + (−0.984 − 0.173i)5-s + 7-s + (0.866 − 0.5i)11-s + (−0.642 − 0.766i)13-s + (−0.939 + 0.342i)17-s + (−0.939 − 0.342i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s + (−0.984 − 0.173i)35-s − i·37-s + (−0.939 + 0.342i)41-s + (−0.342 − 0.939i)43-s + (−0.766 + 0.642i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)5-s + 7-s + (0.866 − 0.5i)11-s + (−0.642 − 0.766i)13-s + (−0.939 + 0.342i)17-s + (−0.939 − 0.342i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s + (−0.984 − 0.173i)35-s − i·37-s + (−0.939 + 0.342i)41-s + (−0.342 − 0.939i)43-s + (−0.766 + 0.642i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.692 + 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1131837998 + 0.2655315797i\)
\(L(\frac12)\) \(\approx\) \(0.1131837998 + 0.2655315797i\)
\(L(1)\) \(\approx\) \(0.8022476947 - 0.03702839470i\)
\(L(1)\) \(\approx\) \(0.8022476947 - 0.03702839470i\)

Euler product

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

−19.101568584361182282254023544154, −18.13254755540404884924000062367, −17.8008746163274808835001155676, −16.73282420034824121530839699185, −16.330737670248928701796287259453, −15.22387112884623231655607115921, −14.841629417405321083814091774230, −14.2460622741129009573313087902, −13.37551774353267773431567131438, −12.34100111287356177269192988591, −11.69930063930031200005453989945, −11.387335521284617951455574288131, −10.52322509818974466234744115979, −9.49121916066680336943192907010, −8.85091951493896856229181571086, −8.05422100028037182374894004609, −7.23708075538472717715995590294, −6.87111222870773649449703115475, −5.68767922619990237560209629999, −4.63473501580731801563200339160, −4.27908361901041556693529763107, −3.43943763955004969461430387121, −2.17109540900998388883315610598, −1.59934405653799690589444666052, −0.09341894459212553965743548238, 1.13219984884039787602960250178, 2.05313720555654265146217951422, 3.16309742036520075565184866315, 4.00454384917868494801713639880, 4.62146311353415204278947996114, 5.41154531189592056414525783559, 6.41755463843896318993145604514, 7.23600157098053732668406712222, 8.06398097040063210763342633862, 8.45806457302763422772414819944, 9.27435733020043224336227194016, 10.38885097802018628546543281203, 10.98794975744762361302180206352, 11.82529966753477887445453606163, 12.086592920374430451797388051096, 13.13619883387020568877729124648, 13.90459662602280544095794594437, 14.833348667532264795920859316207, 15.0897550238022644311178961675, 15.96440076613293344650090578583, 16.77846687884461529639464814984, 17.36335137378610496414304892486, 18.09519646909399180794919831438, 18.87432307899374778393715122083, 19.646745149032797654620299534013

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