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Properties

Label 2-3240-5.4-c1-0-23
Degree 22
Conductor 32403240
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 2i)5-s − 2i·7-s + 5·11-s + 4i·13-s + 6i·17-s + 7·19-s + 4i·23-s + (−3 + 4i)25-s − 5·29-s − 3·31-s + (−4 + 2i)35-s + 2i·37-s − 7·41-s + 6i·43-s + 6i·47-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s − 0.755i·7-s + 1.50·11-s + 1.10i·13-s + 1.45i·17-s + 1.60·19-s + 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.928·29-s − 0.538·31-s + (−0.676 + 0.338i)35-s + 0.328i·37-s − 1.09·41-s + 0.914i·43-s + 0.875i·47-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(649,⋅)\chi_{3240} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 0.894−0.447i)(2,\ 3240,\ (\ :1/2),\ 0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.7314525271.731452527
L(12)L(\frac12) ≈\approx 1.7314525271.731452527
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
5 1+(1+2i)T 1 + (1 + 2i)T
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1−5T+11T2 1 - 5T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1−6iT−17T2 1 - 6iT - 17T^{2}
19 1−7T+19T2 1 - 7T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+7T+41T2 1 + 7T + 41T^{2}
43 1−6iT−43T2 1 - 6iT - 43T^{2}
47 1−6iT−47T2 1 - 6iT - 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1+15T+59T2 1 + 15T + 59T^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1−5T+71T2 1 - 5T + 71T^{2}
73 1−14iT−73T2 1 - 14iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1−3T+89T2 1 - 3T + 89T^{2}
97 1−2iT−97T2 1 - 2iT - 97T^{2}
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   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

−8.751247200358200256677502074403, −7.973103881985362746939187645620, −7.23408295871254035710001612488, −6.56678118655913522222147313566, −5.64194709881183865967712858245, −4.75515824880593334908041869324, −3.74976450356557121483933309561, −3.72212784264987148652800031930, −1.68918136824575440729859233700, −1.15113616788828396308145435499, 0.61838830697776382225402053268, 2.10778927255725965753624548073, 3.14899033437529600440979433568, 3.59387681436439858374163140986, 4.82508550326625055024023385626, 5.61749562875058607116179341784, 6.37484696475546100397345377836, 7.25816312128100823202082167354, 7.59925169840168080336403985486, 8.739766977327316168551549030044

Graph of the ZZ-function along the critical line