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Properties

Label 1-2736-2736.83-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.495 - 0.868i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.495 - 0.868i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.495 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6148681753 - 1.058479331i\)
\(L(\frac12)\) \(\approx\) \(0.6148681753 - 1.058479331i\)
\(L(1)\) \(\approx\) \(0.9941105807 - 0.2720779933i\)
\(L(1)\) \(\approx\) \(0.9941105807 - 0.2720779933i\)

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

−19.40573523972215351704618266631, −18.56347670215397927141731163891, −18.26597596909236884604243380721, −17.46920072661623666505836780932, −16.7532690462867077277579264958, −15.7886872140460418330636108974, −15.3481454740879417658395672310, −14.5892386791565315705896676804, −13.69452664743781627729229431315, −13.1991827598744797042193573950, −12.41785827357734255302306114156, −11.752820938570267779874773378152, −10.566668176082961408678115761736, −10.24723366429573435116862627651, −9.60421429022589126226189927909, −8.41617056704263065619798463807, −8.17711354995741855141378214769, −6.92676870957521476026646993614, −6.12845179703161749765852479650, −5.66741849938033633797688799285, −4.98194073660657588572530960425, −3.58631383628899921117665443067, −2.88356740215050531493886587180, −2.31898782329993883210715599456, −1.201158677622040982436887334713, 0.376332835884815998965760217817, 1.5147977681180798706518158425, 2.32076517300445929434222690636, 3.23313151924675926438785301798, 4.36525344411535005463150167380, 4.83973338222875080352908026666, 5.843941657988308250224871558846, 6.530535628868804358565175254358, 7.35552549459370283576634759737, 8.04627092606615902536982808945, 9.134175737942481017606084191732, 9.69246159248931719234750123711, 10.23534820232664490285289549741, 11.008039024215357201216998946476, 12.10849137877394756217961981091, 12.59595276184534654183379073573, 13.61422196080360616146994071184, 13.78070401286237558087363885547, 14.591279348755496737307154246859, 15.75039981150616294766664993853, 16.31921824667514739606915644110, 16.793333485937532901158907936750, 17.68388312831330424384612108795, 18.17584877185838652323472093162, 19.00564606372438374414178680204

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