↑

Properties

Label 2-40e2-40.29-c1-0-26
Degree 22
Conductor 16001600
Sign −0.948−0.316i-0.948 - 0.316i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 6i·11-s + 6i·17-s − 2i·19-s + 4·27-s + 12i·33-s − 6·41-s − 10·43-s + 7·49-s − 12i·51-s + 4i·57-s + 6i·59-s − 14·67-s + 2i·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.333·9-s − 1.80i·11-s + 1.45i·17-s − 0.458i·19-s + 0.769·27-s + 2.08i·33-s − 0.937·41-s − 1.52·43-s + 49-s − 1.68i·51-s + 0.529i·57-s + 0.781i·59-s − 1.71·67-s + 0.234i·73-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.948−0.316i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((−0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.948−0.316i-0.948 - 0.316i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(1249,⋅)\chi_{1600} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1600, ( :1/2), −0.948−0.316i)(2,\ 1600,\ (\ :1/2),\ -0.948 - 0.316i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1 1
good3 1+2T+3T2 1 + 2T + 3T^{2}
7 1−7T2 1 - 7T^{2}
11 1+6iT−11T2 1 + 6iT - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−6iT−17T2 1 - 6iT - 17T^{2}
19 1+2iT−19T2 1 + 2iT - 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+10T+43T2 1 + 10T + 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+14T+67T2 1 + 14T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−2iT−73T2 1 - 2iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+18T+83T2 1 + 18T + 83T^{2}
89 1+18T+89T2 1 + 18T + 89T^{2}
97 1+10iT−97T2 1 + 10iT - 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

−8.632998009671044337676015389991, −8.493264023253021968128737306529, −7.19053170722770689998093942716, −6.21806256559437420914353141876, −5.86527169376354659790671520497, −5.02968443177532664818177209995, −3.90871129395749100222680498506, −2.93801831066259447993989928945, −1.29213611856867358002892438412, 0, 1.60047304650876806231188647670, 2.86502747756074582961215094783, 4.29779990415778891390719514373, 4.98450710694390351225680959151, 5.64006864436831248426242047187, 6.78432884768124675850943487843, 7.12083300677956764268778798250, 8.167837006198653615517623175890, 9.269512082871218195265542415710

Graph of the ZZ-function along the critical line