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Properties

Label 1-2736-2736.331-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.0542 + 0.998i$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0542 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0542 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.0542 + 0.998i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.0542 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6498107675 + 0.6860612551i\)
\(L(\frac12)\) \(\approx\) \(0.6498107675 + 0.6860612551i\)
\(L(1)\) \(\approx\) \(0.8730604850 + 0.1267649120i\)
\(L(1)\) \(\approx\) \(0.8730604850 + 0.1267649120i\)

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 \( 1 + (0.5 - 0.866i)T \)
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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

−19.23071393270087079871372452411, −18.3545676251755056698559310021, −17.74797603257954332877628203733, −16.81365475700506484590009728289, −16.22510903047677417169638843174, −15.60153199294957667822661526350, −15.13954356510168821370275813975, −13.88747173167158555148747505489, −13.207335594355232942831341182823, −12.887294599149505662457291119098, −11.894391817449088809343564204136, −11.38901251307580003370786173967, −10.420073582718544352088946577611, −9.522583083438165966752335636324, −8.95618138564240059751770991983, −8.268224120263323768808052547214, −7.67843447469764678810712342293, −6.257843182358475839158763193323, −5.93760218472061183587990964450, −5.13515021300195622321664395153, −4.17679677012714803522260173612, −3.46001509662434734029811400933, −2.33346781873721146113159477657, −1.630740940285817366582068385511, −0.34259966162306503783483935220, 0.92379784091861290658968952558, 2.244754236863868468005680766, 2.94162011109898478442603605110, 3.709687962931863314938827452914, 4.52500181799359426008763227893, 5.55672352954132521935206026426, 6.37046255777672993017101999010, 7.1344068982601815953542167053, 7.552425026174806767113712415385, 8.54430403397589369023658362514, 9.503418429296908628832750344963, 10.30139892676410074360450817388, 10.75027795889856679129303639798, 11.29984820689848861663690018951, 12.54640093065006094095229610060, 12.99979215786357471327702117443, 13.92011918993509714984197510895, 14.32535347059805116345880504793, 15.26686899630107476346612596546, 16.024455884339644516562936357064, 16.35023997946850477662433512321, 17.64489440853644858299094159293, 18.04991168662720945920052092644, 18.536516859338588841923100172324, 19.54833823178803112041310400158

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