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Properties

Label 1-2736-2736.259-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.600 - 0.799i$
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 − i·13-s + (−0.5 + 0.866i)17-s + 23-s + (0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)35-s − i·37-s + (−0.5 − 0.866i)41-s + i·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 − i·13-s + (−0.5 + 0.866i)17-s + 23-s + (0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)35-s − i·37-s + (−0.5 − 0.866i)41-s + i·43-s + (0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063733331 - 0.5316196951i\)
\(L(\frac12)\) \(\approx\) \(1.063733331 - 0.5316196951i\)
\(L(1)\) \(\approx\) \(0.8983585593 - 0.09875622172i\)
\(L(1)\) \(\approx\) \(0.8983585593 - 0.09875622172i\)

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 \( 1 - iT \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
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 \( 1 + iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−19.41565866731367639043168169872, −18.75244268646936531749496807202, −18.11926728094890292047896624912, −16.971559491881565487529558473722, −16.65421454237860218917503114623, −15.65491071078480045312019183381, −15.38166976682744888229823216253, −14.488987690110353500402155417302, −13.68594981892187950149275776820, −12.77332213812614856553975583596, −12.17874722800031536225371383469, −11.62444423798268619761060192793, −11.039021157952858712019972078115, −9.73885624705687762143713148131, −9.093824701364379224864215231664, −8.80276276713499994758344876025, −7.66573892781983079554435988988, −6.947167679647487151144932605505, −6.29278709129038043317072197282, −5.230699444201008721685369705488, −4.485006002238344423387840415943, −3.84225291734569103284985165592, −2.831891660836965131401052477256, −1.96876729237691823306373441506, −0.82866665510045867455324339829, 0.521202132912577649593633133889, 1.445462129873609536587209076191, 2.95642933799581906422194874990, 3.40185515631865055209838766717, 4.129228438303778234333653633647, 4.99777776533946360954166816082, 6.17310651744246186261706747730, 6.78131045984114170523494693687, 7.380620385627107333463154450836, 8.319682438268475807587337949233, 8.8524677282335125213386627813, 10.00091682586622967955907744182, 10.67676489425753262482769682751, 11.07685171727962104206368382273, 12.07497902869954471790332819755, 12.6933396218671246559012072643, 13.522632254152669309257860057522, 14.20465142723449719345398073598, 15.059225221536036471767381268759, 15.484787977182917916337779323110, 16.445961233406794165295012400941, 16.92753271713822354952455664923, 17.73860454895766349714784954077, 18.513804663451191853183517235980, 19.46876726902511077361409028570

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