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Properties

Label 1-2736-2736.293-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: $\chi_{2736} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.600 + 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.141082129 + 1.070043962i\)
\(L(\frac12)\) \(\approx\) \(2.141082129 + 1.070043962i\)
\(L(1)\) \(\approx\) \(1.412681158 + 0.2491096997i\)
\(L(1)\) \(\approx\) \(1.412681158 + 0.2491096997i\)

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

−18.94973686830724899572171376719, −18.473449465148778049093741782386, −17.73605841313103955681079337965, −16.944437123690165320541801605428, −16.65063919662635507075422128038, −15.39878453745800727802376017387, −15.07215875762772569938812401035, −14.06086983430128050638999983696, −13.577735405203183788082709628683, −12.72380650737265731875366915933, −12.05309229775171491521022777210, −11.364420704021516357429499874699, −10.55049821519559505911490335177, −9.49792804567022087693470816905, −9.225086490859107553579314733669, −8.33726776927164696501303043938, −7.66784772016970959520330813345, −6.499176100312924757684335105107, −5.82672578322720800532451640328, −5.239403504101400598576391789157, −4.52078496532125260045766463326, −3.26383963146490495605155749360, −2.57464979340075472972296664228, −1.59292094373022913711831520463, −0.78204221923722935309016224187, 1.35038007666047828556598729564, 1.63447300795455926946655095676, 2.81136465922056966328625598192, 3.80953202686632341032615543006, 4.45157666554247138689763445795, 5.394681114357773031507449492602, 6.285909162514000483361637946141, 6.9732255789837461351154548311, 7.47623287669155794158506974890, 8.64766817699628605270215593292, 9.35310246193878805220027102904, 9.98282571126970920690426369174, 10.88658910757941953587913423028, 11.21344950513137045850084464922, 12.343345725936376771372123873616, 12.98617199630110952929954177151, 13.91341265599782609379445329863, 14.45279440259166185073422710813, 14.71940413346030901855540271488, 15.91807708546783240176871521290, 16.85907395994458407812912032856, 17.23078039271460696020626810329, 17.75590042561893145897780290554, 18.721952739337247399144798736485, 19.294509160834635433687537677115

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