↑

Properties

Label 2-2664-296.147-c0-0-3
Degree 22
Conductor 26642664
Sign 0.382+0.923i0.382 + 0.923i
Analytic cond. 1.329501.32950
Root an. cond. 1.153041.15304
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  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s − 1.84·5-s + 1.41i·7-s + (−0.923 + 0.382i)8-s + (−0.707 + 1.70i)10-s + (1.30 + 0.541i)14-s + i·16-s − 1.84i·17-s + (1.30 + 1.30i)20-s + 1.84·23-s + 2.41·25-s + (1.00 − i)28-s + 0.765·29-s + (0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s − 1.84·5-s + 1.41i·7-s + (−0.923 + 0.382i)8-s + (−0.707 + 1.70i)10-s + (1.30 + 0.541i)14-s + i·16-s − 1.84i·17-s + (1.30 + 1.30i)20-s + 1.84·23-s + 2.41·25-s + (1.00 − i)28-s + 0.765·29-s + (0.923 + 0.382i)32-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: 0.382+0.923i0.382 + 0.923i
Analytic conductor: 1.329501.32950
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(739,⋅)\chi_{2664} (739, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :0), 0.382+0.923i)(2,\ 2664,\ (\ :0),\ 0.382 + 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.89988238180.8998823818
L(12)L(\frac12) ≈\approx 0.89988238180.8998823818
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+(−0.382+0.923i)T 1 + (-0.382 + 0.923i)T
3 1 1
37 1+iT 1 + iT
good5 1+1.84T+T2 1 + 1.84T + T^{2}
7 1−1.41iT−T2 1 - 1.41iT - T^{2}
11 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
17 1+1.84iT−T2 1 + 1.84iT - T^{2}
19 1−T2 1 - T^{2}
23 1−1.84T+T2 1 - 1.84T + T^{2}
29 1−0.765T+T2 1 - 0.765T + T^{2}
31 1+T2 1 + T^{2}
41 1+T2 1 + T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1−1.84iT−T2 1 - 1.84iT - T^{2}
61 1+T2 1 + T^{2}
67 1−1.41T+T2 1 - 1.41T + T^{2}
71 1−T2 1 - T^{2}
73 1−1.41T+T2 1 - 1.41T + T^{2}
79 1+T2 1 + T^{2}
83 1+T2 1 + T^{2}
89 1−0.765iT−T2 1 - 0.765iT - 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

−8.977106575892864554128796470041, −8.377680652823940299448848468934, −7.43168210038482305837555609867, −6.65024640637757604134622637285, −5.32804851592166107262293779607, −4.93571787195976445387616057831, −4.01637560996657393869427880899, −3.03086757903055465057292838216, −2.59298913796615371330345718112, −0.795240026356658582282098779273, 0.874008472681171131240413780436, 3.24879307978255272710451852445, 3.73357428342909657581953319407, 4.42916645294775167487959775390, 5.04101526718071522720207282624, 6.47996042471081427569097084955, 6.91517655594228027672360788495, 7.64447584027773166729967179063, 8.203909973594439901147148697282, 8.677529477118951821468296493547

Graph of the ZZ-function along the critical line