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Properties

Label 1-2736-2736.619-r1-0-0
Degree $1$
Conductor $2736$
Sign $0.347 + 0.937i$
Analytic cond. $294.024$
Root an. cond. $294.024$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.347 + 0.937i$
Analytic conductor: \(294.024\)
Root analytic conductor: \(294.024\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (1:\ ),\ 0.347 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4655271571 + 0.3241260662i\)
\(L(\frac12)\) \(\approx\) \(0.4655271571 + 0.3241260662i\)
\(L(1)\) \(\approx\) \(0.7388425733 + 0.02507287561i\)
\(L(1)\) \(\approx\) \(0.7388425733 + 0.02507287561i\)

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

−19.21935454443581385159108746901, −18.05740760932167234924085639302, −17.66195474451956575006504909509, −16.74398163798009884458042088397, −16.05434831605545683453244564978, −15.56392678800171561925825867277, −14.882584687910438571100946574773, −13.82846716737733947875695276948, −13.11787113273985327318752876542, −12.49826297182259660824166261870, −12.105899058474277982388617720972, −11.05556938487203558649726066439, −10.234663476407261032864666289784, −9.51096617182374857136590677232, −8.7331884631772128756758814894, −8.23707711330407064312823795503, −7.39201029808429182870213114769, −6.3335310972056288895051465103, −5.44048484617255849521605446693, −5.19542125315318515067138443947, −4.07312721550507658699134551587, −3.13272300311081347093374936460, −2.31748644074127904702904272405, −1.391585665335288124632992627439, −0.1763810305934052386587378440, 0.442070662781730414537499015254, 2.02849733737666708613518390387, 2.6087154108343085170303715390, 3.449189202437855330158564709703, 4.4281892425801609004561213926, 5.00492170311721420858864134718, 6.46563036548423918185945270896, 6.58358164444084818605319604448, 7.63759876700830204620080768432, 7.97638042889001180458922396415, 9.49018315900902216165413021311, 9.86361076250272073485629102193, 10.518408019464331958123861799669, 11.30432871176244619110724838409, 11.99235955735158962756195095167, 12.95895795177923224640021651705, 13.60941462616716822037924318673, 14.22248612764109417726086491666, 14.967770237843902268047034983187, 15.68439820708691892006388514633, 16.34062836612559733908669629938, 17.26827180696423533025453674963, 17.72220941766387808793105734915, 18.768025749205504044413139016603, 18.95032063340633689333696600494

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