↑

Properties

Label 2-2640-5.4-c1-0-47
Degree 22
Conductor 26402640
Sign 0.241+0.970i0.241 + 0.970i
Analytic cond. 21.080521.0805
Root an. cond. 4.591354.59135
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  − i·3-s + (2.17 − 0.539i)5-s − 0.290i·7-s − 9-s + 11-s − 6.97i·13-s + (−0.539 − 2.17i)15-s + 4.78i·17-s + 7.75·19-s − 0.290·21-s + 4i·23-s + (4.41 − 2.34i)25-s + i·27-s + 7.41·29-s − 6.34·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.970 − 0.241i)5-s − 0.109i·7-s − 0.333·9-s + 0.301·11-s − 1.93i·13-s + (−0.139 − 0.560i)15-s + 1.16i·17-s + 1.77·19-s − 0.0634·21-s + 0.834i·23-s + (0.883 − 0.468i)25-s + 0.192i·27-s + 1.37·29-s − 1.13·31-s + ⋯

Functional equation

Λ(s)=(2640s/2 ΓC(s) L(s)=((0.241+0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2640s/2 ΓC(s+1/2) L(s)=((0.241+0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26402640    =    24⋅3⋅5⋅112^{4} \cdot 3 \cdot 5 \cdot 11
Sign: 0.241+0.970i0.241 + 0.970i
Analytic conductor: 21.080521.0805
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2640(529,⋅)\chi_{2640} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2640, ( :1/2), 0.241+0.970i)(2,\ 2640,\ (\ :1/2),\ 0.241 + 0.970i)

Particular Values

L(1)L(1) ≈\approx 2.3023061772.302306177
L(12)L(\frac12) ≈\approx 2.3023061772.302306177
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+iT 1 + iT
5 1+(−2.17+0.539i)T 1 + (-2.17 + 0.539i)T
11 1−T 1 - T
good7 1+0.290iT−7T2 1 + 0.290iT - 7T^{2}
13 1+6.97iT−13T2 1 + 6.97iT - 13T^{2}
17 1−4.78iT−17T2 1 - 4.78iT - 17T^{2}
19 1−7.75T+19T2 1 - 7.75T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1−7.41T+29T2 1 - 7.41T + 29T^{2}
31 1+6.34T+31T2 1 + 6.34T + 31T^{2}
37 1+3.41iT−37T2 1 + 3.41iT - 37T^{2}
41 1+7.41T+41T2 1 + 7.41T + 41T^{2}
43 1−0.290iT−43T2 1 - 0.290iT - 43T^{2}
47 1+5.26iT−47T2 1 + 5.26iT - 47T^{2}
53 1+5.75iT−53T2 1 + 5.75iT - 53T^{2}
59 1−3.60T+59T2 1 - 3.60T + 59T^{2}
61 1+6.68T+61T2 1 + 6.68T + 61T^{2}
67 1+6.15iT−67T2 1 + 6.15iT - 67T^{2}
71 1−5.07T+71T2 1 - 5.07T + 71T^{2}
73 1−1.12iT−73T2 1 - 1.12iT - 73T^{2}
79 1+0.921T+79T2 1 + 0.921T + 79T^{2}
83 1−1.70iT−83T2 1 - 1.70iT - 83T^{2}
89 1+4.34T+89T2 1 + 4.34T + 89T^{2}
97 1−4.68iT−97T2 1 - 4.68iT - 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

−8.612426118961096915249678971397, −7.951557023014428278439246715571, −7.21571676879084708649239357116, −6.32900401284163195807664710552, −5.48813961008071820778141505275, −5.23238595963172697069376454004, −3.65507327557211832922316719412, −2.90823864962864757402690948332, −1.72359574976105642656954148587, −0.841842234192493177452283137314, 1.28855809257500958653215153678, 2.42771267176284252716214710961, 3.25956660427373176245916677486, 4.42885752643919455481606105279, 5.04494629396333221717947691578, 5.90651269742790002367478221782, 6.78019996983690098574598174258, 7.23786660927260251281226627345, 8.572906420334376669082415727836, 9.238236670983916725768442536904

Graph of the ZZ-function along the critical line