↑

Properties

Label 2-42e2-7.2-c3-0-11
Degree 22
Conductor 17641764
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 104.079104.079
Root an. cond. 10.201910.2019
Motivic weight 33
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 − 5.19i)5-s + (−6 − 10.3i)11-s − 82·13-s + (−15 − 25.9i)17-s + (−34 + 58.8i)19-s + (108 − 187. i)23-s + (44.5 + 77.0i)25-s − 246·29-s + (56 + 96.9i)31-s + (−55 + 95.2i)37-s + 246·41-s − 172·43-s + (96 − 166. i)47-s + (279 + 483. i)53-s − 72·55-s + ⋯
L(s)  = 1  + (0.268 − 0.464i)5-s + (−0.164 − 0.284i)11-s − 1.74·13-s + (−0.214 − 0.370i)17-s + (−0.410 + 0.711i)19-s + (0.979 − 1.69i)23-s + (0.355 + 0.616i)25-s − 1.57·29-s + (0.324 + 0.561i)31-s + (−0.244 + 0.423i)37-s + 0.937·41-s − 0.609·43-s + (0.297 − 0.516i)47-s + (0.723 + 1.25i)53-s − 0.176·55-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+3/2) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 104.079104.079
Root analytic conductor: 10.201910.2019
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1764(1549,⋅)\chi_{1764} (1549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :3/2), 0.605−0.795i)(2,\ 1764,\ (\ :3/2),\ 0.605 - 0.795i)

Particular Values

L(2)L(2) ≈\approx 1.3063552751.306355275
L(12)L(\frac12) ≈\approx 1.3063552751.306355275
L(52)L(\frac{5}{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
3 1 1
7 1 1
good5 1+(−3+5.19i)T+(−62.5−108.i)T2 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2}
11 1+(6+10.3i)T+(−665.5+1.15e3i)T2 1 + (6 + 10.3i)T + (-665.5 + 1.15e3i)T^{2}
13 1+82T+2.19e3T2 1 + 82T + 2.19e3T^{2}
17 1+(15+25.9i)T+(−2.45e3+4.25e3i)T2 1 + (15 + 25.9i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(34−58.8i)T+(−3.42e3−5.94e3i)T2 1 + (34 - 58.8i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−108+187.i)T+(−6.08e3−1.05e4i)T2 1 + (-108 + 187. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+246T+2.43e4T2 1 + 246T + 2.43e4T^{2}
31 1+(−56−96.9i)T+(−1.48e4+2.57e4i)T2 1 + (-56 - 96.9i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(55−95.2i)T+(−2.53e4−4.38e4i)T2 1 + (55 - 95.2i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−246T+6.89e4T2 1 - 246T + 6.89e4T^{2}
43 1+172T+7.95e4T2 1 + 172T + 7.95e4T^{2}
47 1+(−96+166.i)T+(−5.19e4−8.99e4i)T2 1 + (-96 + 166. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−279−483.i)T+(−7.44e4+1.28e5i)T2 1 + (-279 - 483. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−270−467.i)T+(−1.02e5+1.77e5i)T2 1 + (-270 - 467. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(55−95.2i)T+(−1.13e5−1.96e5i)T2 1 + (55 - 95.2i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(70+121.i)T+(−1.50e5+2.60e5i)T2 1 + (70 + 121. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−840T+3.57e5T2 1 - 840T + 3.57e5T^{2}
73 1+(−275−476.i)T+(−1.94e5+3.36e5i)T2 1 + (-275 - 476. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−104+180.i)T+(−2.46e5−4.26e5i)T2 1 + (-104 + 180. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+516T+5.71e5T2 1 + 516T + 5.71e5T^{2}
89 1+(699−1.21e3i)T+(−3.52e5−6.10e5i)T2 1 + (699 - 1.21e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−1.58e3T+9.12e5T2 1 - 1.58e3T + 9.12e5T^{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

−9.052221382273805301408406346154, −8.375699390839973244842821999413, −7.37959852258347167941793365177, −6.81063842696461118897875355416, −5.66017500480244854917819616112, −5.01126581062462448500396139857, −4.23005774667191642212710346574, −2.93537256464650623688272063301, −2.12033042530444823069714710582, −0.802563063807657712302807523489, 0.34269278439099835618194304101, 1.94108027749357494432653834038, 2.64690620064750497691621759005, 3.74940038934871434895651144395, 4.83991701422585894997080783907, 5.46909049364320176268653737552, 6.56478314621434743514068378396, 7.26146181736350600957438556331, 7.82018216569863322577176746571, 9.020422165924869893164370914615

Graph of the ZZ-function along the critical line