↑

Properties

Label 2-2736-228.227-c0-0-10
Degree 22
Conductor 27362736
Sign −0.908+0.418i-0.908 + 0.418i
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.93i·5-s + i·7-s − 1.93·11-s − 0.517i·17-s − i·19-s − 1.41·23-s − 2.73·25-s + 1.93·35-s − 1.73i·43-s + 0.517·47-s + 3.73i·55-s − 1.73·61-s − 73-s − 1.93i·77-s + 1.41·83-s + ⋯
L(s)  = 1  − 1.93i·5-s + i·7-s − 1.93·11-s − 0.517i·17-s − i·19-s − 1.41·23-s − 2.73·25-s + 1.93·35-s − 1.73i·43-s + 0.517·47-s + 3.73i·55-s − 1.73·61-s − 73-s − 1.93i·77-s + 1.41·83-s + ⋯

Functional equation

Λ(s)=(2736s/2 ΓC(s) L(s)=((−0.908+0.418i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2736s/2 ΓC(s) L(s)=((−0.908+0.418i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27362736    =    24⋅32⋅192^{4} \cdot 3^{2} \cdot 19
Sign: −0.908+0.418i-0.908 + 0.418i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2736(2735,⋅)\chi_{2736} (2735, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2736, ( :0), −0.908+0.418i)(2,\ 2736,\ (\ :0),\ -0.908 + 0.418i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.57898433800.5789843380
L(12)L(\frac12) ≈\approx 0.57898433800.5789843380
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
19 1+iT 1 + iT
good5 1+1.93iT−T2 1 + 1.93iT - T^{2}
7 1−iT−T2 1 - iT - T^{2}
11 1+1.93T+T2 1 + 1.93T + T^{2}
13 1−T2 1 - T^{2}
17 1+0.517iT−T2 1 + 0.517iT - T^{2}
23 1+1.41T+T2 1 + 1.41T + T^{2}
29 1+T2 1 + T^{2}
31 1+T2 1 + T^{2}
37 1−T2 1 - T^{2}
41 1+T2 1 + T^{2}
43 1+1.73iT−T2 1 + 1.73iT - T^{2}
47 1−0.517T+T2 1 - 0.517T + T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1+1.73T+T2 1 + 1.73T + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+T2 1 + T^{2}
83 1−1.41T+T2 1 - 1.41T + T^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.768023651352264485095572631345, −8.035384069926729763254416096496, −7.49932610770339486630728059281, −6.06273402715115050056194350758, −5.32645538800995296924400381764, −5.04559306447479934794994027294, −4.12910019873926478865732751075, −2.71157440457987262398246981670, −1.92800372331885536156797424444, −0.33621524460988506890920204207, 1.97556073725916873941508651807, 2.87296423400372615319595313313, 3.59084606235859465546508087826, 4.47160297415081273664609038226, 5.79389342278027564452531630299, 6.22992941158733707669985663770, 7.24849135833409535718776060620, 7.71415707032317004086231541543, 8.161680858577362267074985390650, 9.735021410767016169660010183776

Graph of the ZZ-function along the critical line