↑

Properties

Label 2-4788-133.132-c1-0-11
Degree 22
Conductor 47884788
Sign −0.997−0.0655i-0.997 - 0.0655i
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  + 1.58i·5-s + (0.618 + 2.57i)7-s − 1.61·11-s + 3.38·13-s + 2.57i·17-s + (−1.29 + 4.16i)19-s − 7.47·23-s + 2.47·25-s + 8.71i·29-s − 8.06·31-s + (−4.09 + 0.982i)35-s − 5.38i·37-s + 8.06·41-s − 5.70·43-s − 4.76i·47-s + ⋯
L(s)  = 1  + 0.711i·5-s + (0.233 + 0.972i)7-s − 0.487·11-s + 0.939·13-s + 0.623i·17-s + (−0.296 + 0.954i)19-s − 1.55·23-s + 0.494·25-s + 1.61i·29-s − 1.44·31-s + (−0.691 + 0.166i)35-s − 0.885i·37-s + 1.25·41-s − 0.870·43-s − 0.695i·47-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=((−0.997−0.0655i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=((−0.997−0.0655i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: −0.997−0.0655i-0.997 - 0.0655i
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4788(3457,⋅)\chi_{4788} (3457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), −0.997−0.0655i)(2,\ 4788,\ (\ :1/2),\ -0.997 - 0.0655i)

Particular Values

L(1)L(1) ≈\approx 1.0264905981.026490598
L(12)L(\frac12) ≈\approx 1.0264905981.026490598
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
3 1 1
7 1+(−0.618−2.57i)T 1 + (-0.618 - 2.57i)T
19 1+(1.29−4.16i)T 1 + (1.29 - 4.16i)T
good5 1−1.58iT−5T2 1 - 1.58iT - 5T^{2}
11 1+1.61T+11T2 1 + 1.61T + 11T^{2}
13 1−3.38T+13T2 1 - 3.38T + 13T^{2}
17 1−2.57iT−17T2 1 - 2.57iT - 17T^{2}
23 1+7.47T+23T2 1 + 7.47T + 23T^{2}
29 1−8.71iT−29T2 1 - 8.71iT - 29T^{2}
31 1+8.06T+31T2 1 + 8.06T + 31T^{2}
37 1+5.38iT−37T2 1 + 5.38iT - 37T^{2}
41 1−8.06T+41T2 1 - 8.06T + 41T^{2}
43 1+5.70T+43T2 1 + 5.70T + 43T^{2}
47 1+4.76iT−47T2 1 + 4.76iT - 47T^{2}
53 1+14.0iT−53T2 1 + 14.0iT - 53T^{2}
59 1+0.799T+59T2 1 + 0.799T + 59T^{2}
61 1−6.73iT−61T2 1 - 6.73iT - 61T^{2}
67 1+2.05iT−67T2 1 + 2.05iT - 67T^{2}
71 1−12.0iT−71T2 1 - 12.0iT - 71T^{2}
73 1+9.30iT−73T2 1 + 9.30iT - 73T^{2}
79 1−10.7iT−79T2 1 - 10.7iT - 79T^{2}
83 1+2.94iT−83T2 1 + 2.94iT - 83T^{2}
89 1−8.37T+89T2 1 - 8.37T + 89T^{2}
97 1+10.1T+97T2 1 + 10.1T + 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

−8.532487966313609442604740789197, −8.048621440367089008238112666359, −7.18562782552944775968411897293, −6.36016346634099301421136557927, −5.76476246758700158283968935850, −5.18384638662739013433498689293, −3.92628105030431241802483463539, −3.39859330235371063147980647975, −2.31055602034985936782372870263, −1.62239925364035530825913660853, 0.27844887143673088651115118446, 1.25168890381352421813605677205, 2.34800203479382137558233726750, 3.46189540712130906559859172605, 4.33199453649359158257995877567, 4.77206700408021423881445280411, 5.78494969188700799070515326874, 6.41743371465637730950104282349, 7.39735134761185402760823617102, 7.88778845291661389928392401598

Graph of the ZZ-function along the critical line