↑

Properties

Label 2-3864-3864.275-c0-0-7
Degree 22
Conductor 38643864
Sign −0.0633−0.997i-0.0633 - 0.997i
Analytic cond. 1.928381.92838
Root an. cond. 1.388661.38866
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

Λ(s)=(3864s/2 ΓC(s) L(s)=((−0.0633−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3864s/2 ΓC(s) L(s)=((−0.0633−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38643864    =    23⋅3⋅7⋅232^{3} \cdot 3 \cdot 7 \cdot 23
Sign: −0.0633−0.997i-0.0633 - 0.997i
Analytic conductor: 1.928381.92838
Root analytic conductor: 1.388661.38866
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3864(275,⋅)\chi_{3864} (275, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3864, ( :0), −0.0633−0.997i)(2,\ 3864,\ (\ :0),\ -0.0633 - 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.63508574800.6350857480
L(12)L(\frac12) ≈\approx 0.63508574800.6350857480
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
23 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1−T+T2 1 - T + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T+T2 1 - T + T^{2}
73 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+2T+T2 1 + 2T + T^{2}
89 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)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

−8.219189188642780404542904303302, −7.16938022147280820033971489848, −6.50210872235113573944721055360, −5.95114725925289441576948395211, −4.94359325664082042010431851080, −4.40462312584746520102431715530, −3.11991575713817529575914785956, −2.64305245509028095759195578890, −1.31962355013442583721465561600, −0.33066867244819681763962404691, 2.22772981036653817504995239704, 3.36646395070256590935335234095, 3.93190234913587210165587762504, 5.05906572231696057694558227822, 5.27982402753625588314713162915, 6.19419868316913839343937281182, 6.74486036309056080591680124825, 7.65040822581161830897151123695, 8.504212232770323857713115624242, 9.209634382667069929620020228269

Graph of the ZZ-function along the critical line