↑

Properties

Label 1-2736-2736.1379-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.908 + 0.417i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·5-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)23-s − 25-s − i·29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)35-s − i·37-s + 41-s + (−0.866 − 0.5i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  − i·5-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)23-s − 25-s − i·29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)35-s − i·37-s + 41-s + (−0.866 − 0.5i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.908 + 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.430901237 + 0.3132375497i\)
\(L(\frac12)\) \(\approx\) \(1.430901237 + 0.3132375497i\)
\(L(1)\) \(\approx\) \(1.016053817 + 0.1089342710i\)
\(L(1)\) \(\approx\) \(1.016053817 + 0.1089342710i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−19.16079365465033018259313732914, −18.416747799796040504818780171542, −18.08570689587152108054903233610, −16.80073082283541254712822867652, −16.38382488441423738884631492757, −15.81768616216258749478496490443, −15.20591607661249394155937286414, −14.04474130108239041014604851494, −13.5355805741600525695605276119, −12.66367035130374594566955610347, −12.28520518621764226208023087163, −11.414505641460984285828643665389, −10.63183377557573776814925816011, −9.68874340710619383121981121983, −8.967061967710679139020840758800, −8.53118324981314067036907939350, −7.73891192337484156282946478274, −6.62752837382939348190268205639, −5.92958506603758553386788030039, −5.13934795250957911560744196065, −4.60410156597306859099250074145, −3.34918978635868414691445728827, −2.7705957048885805409238575968, −1.63746563781155964989743909029, −0.68163701191040203314630650840, 0.73511224370580326878481514360, 1.95449965123361204257478207047, 2.86943782779073489246727559953, 3.63234715030066891877295287834, 4.21737764001587178914746337338, 5.52367985641974081349713510588, 6.09987355202030748711266010478, 6.95680993542195070213438874100, 7.65757909101401995020395681604, 8.15247528357555289691708469653, 9.47160070737379440086830642883, 10.06074425508432283406165697788, 10.71063734144653711484235571473, 11.16396352864748874556246495152, 12.22756946837775901348859869421, 13.17299680289189731283325125149, 13.46778365871549230127797031951, 14.36709523789775816629197373819, 15.179964176072741013895677334669, 15.63987258057778064871525852852, 16.4407567058070570047245443975, 17.515956434064718920042551066904, 17.6651403178942383185055221828, 18.85595868375637727110693629119, 19.0631049142608300316618568445

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