↑

Properties

Label 2-2816-88.43-c1-0-79
Degree 22
Conductor 28162816
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 22.485822.4858
Root an. cond. 4.741924.74192
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.31·3-s − 3i·5-s + 8·9-s + 3.31·11-s − 9.94i·15-s + 3.31i·23-s − 4·25-s + 16.5·27-s + 9.94i·31-s + 11·33-s − 7i·37-s − 24i·45-s − 6.63i·47-s − 7·49-s − 6i·53-s + ⋯
L(s)  = 1  + 1.91·3-s − 1.34i·5-s + 2.66·9-s + 1.00·11-s − 2.56i·15-s + 0.691i·23-s − 0.800·25-s + 3.19·27-s + 1.78i·31-s + 1.91·33-s − 1.15i·37-s − 3.57i·45-s − 0.967i·47-s − 49-s − 0.824i·53-s + ⋯

Functional equation

Λ(s)=(2816s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2816s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28162816    =    28⋅112^{8} \cdot 11
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 22.485822.4858
Root analytic conductor: 4.741924.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2816(1407,⋅)\chi_{2816} (1407, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2816, ( :1/2), 0.707+0.707i)(2,\ 2816,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 4.1642536624.164253662
L(12)L(\frac12) ≈\approx 4.1642536624.164253662
L(32)L(\frac{3}{2}) 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
11 1−3.31T 1 - 3.31T
good3 1−3.31T+3T2 1 - 3.31T + 3T^{2}
5 1+3iT−5T2 1 + 3iT - 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1+13T2 1 + 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1−3.31iT−23T2 1 - 3.31iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−9.94iT−31T2 1 - 9.94iT - 31T^{2}
37 1+7iT−37T2 1 + 7iT - 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+6.63iT−47T2 1 + 6.63iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1+3.31T+59T2 1 + 3.31T + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+9.94T+67T2 1 + 9.94T + 67T^{2}
71 1−16.5iT−71T2 1 - 16.5iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1+17T+97T2 1 + 17T + 97T^{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

−8.594219145927245831589693873961, −8.346137031523468576610632620350, −7.37106032732570703157523247940, −6.74073489459835952980295578983, −5.40332961934681688588666639330, −4.51761137104174719945196121106, −3.84820595234986949428889067284, −3.09277439014469618477807125665, −1.86137532145363574658900962186, −1.21986679994038940261050732527, 1.52562431655728173306011074383, 2.52183389423270904862779536715, 3.08465445247433503966741023318, 3.86754329358391859924914604225, 4.56400322265379211794966593772, 6.28112950243654580729310155764, 6.69706700067729986285344019398, 7.63748514795801563898215215484, 7.973261088010321470903721658248, 9.004101929207985475408220972290

Graph of the ZZ-function along the critical line