↑

Properties

Label 2-40e2-100.11-c0-0-1
Degree 22
Conductor 16001600
Sign −0.187+0.982i-0.187 + 0.982i
Analytic cond. 0.7985040.798504
Root an. cond. 0.8935900.893590
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  + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)9-s + (0.5 + 0.363i)13-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (−1.30 − 0.951i)37-s + (−0.5 − 0.363i)41-s + (−0.309 + 0.951i)45-s + 49-s + (−0.190 + 0.587i)53-s + (0.5 − 0.363i)61-s + (0.190 − 0.587i)65-s + (−0.5 + 0.363i)73-s + (0.309 + 0.951i)81-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)9-s + (0.5 + 0.363i)13-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (−1.30 − 0.951i)37-s + (−0.5 − 0.363i)41-s + (−0.309 + 0.951i)45-s + 49-s + (−0.190 + 0.587i)53-s + (0.5 − 0.363i)61-s + (0.190 − 0.587i)65-s + (−0.5 + 0.363i)73-s + (0.309 + 0.951i)81-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.187+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.187+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.187+0.982i-0.187 + 0.982i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(511,⋅)\chi_{1600} (511, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), −0.187+0.982i)(2,\ 1600,\ (\ :0),\ -0.187 + 0.982i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83911052260.8391105226
L(12)L(\frac12) ≈\approx 0.83911052260.8391105226
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 1
5 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
good3 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
7 1−T2 1 - T^{2}
11 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
13 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
17 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}
19 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
23 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
29 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
31 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
41 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
43 1−T2 1 - T^{2}
47 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
53 1+(0.190−0.587i)T+(−0.809−0.587i)T2 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
67 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
71 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
73 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}
79 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
97 1+(0.5−1.53i)T+(−0.809−0.587i)T2 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)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

−9.060530546209481063756938441076, −8.849292540963149531341311797377, −7.88585154815558867769082933207, −7.00371766543096091699328867620, −6.04995159932846715639649453724, −5.24443509450568258219331981418, −4.37528734040920828418377688983, −3.46109081550391221025307731017, −2.24673950601836737448916783596, −0.65014870029216572411316128899, 1.83983109858557752187143826474, 3.01162637887682082632787829172, 3.71058196575893925915097754418, 4.90580606702507158354179508218, 5.88923826015952475583707126015, 6.57696057727015355967606272153, 7.42126686632240611254181321579, 8.417282979627949200631293199865, 8.674595640950674101332078338257, 10.12576251603664497746291482382

Graph of the ZZ-function along the critical line