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Properties

Label 2-52e2-13.12-c1-0-3
Degree 22
Conductor 27042704
Sign −0.691+0.722i-0.691 + 0.722i
Analytic cond. 21.591521.5915
Root an. cond. 4.646674.64667
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  − 0.801·3-s + 2.80i·5-s + 2.69i·7-s − 2.35·9-s + 1.19i·11-s − 2.24i·15-s − 1.13·17-s + 1.93i·19-s − 2.15i·21-s − 4.60·23-s − 2.85·25-s + 4.29·27-s − 7.89·29-s + 5.89i·31-s − 0.960i·33-s + ⋯
L(s)  = 1  − 0.462·3-s + 1.25i·5-s + 1.01i·7-s − 0.785·9-s + 0.361i·11-s − 0.580i·15-s − 0.275·17-s + 0.444i·19-s − 0.471i·21-s − 0.959·23-s − 0.570·25-s + 0.826·27-s − 1.46·29-s + 1.05i·31-s − 0.167i·33-s + ⋯

Functional equation

Λ(s)=(2704s/2 ΓC(s) L(s)=((−0.691+0.722i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2704s/2 ΓC(s+1/2) L(s)=((−0.691+0.722i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27042704    =    24⋅1322^{4} \cdot 13^{2}
Sign: −0.691+0.722i-0.691 + 0.722i
Analytic conductor: 21.591521.5915
Root analytic conductor: 4.646674.64667
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2704(337,⋅)\chi_{2704} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2704, ( :1/2), −0.691+0.722i)(2,\ 2704,\ (\ :1/2),\ -0.691 + 0.722i)

Particular Values

L(1)L(1) ≈\approx 0.42905410610.4290541061
L(12)L(\frac12) ≈\approx 0.42905410610.4290541061
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
13 1 1
good3 1+0.801T+3T2 1 + 0.801T + 3T^{2}
5 1−2.80iT−5T2 1 - 2.80iT - 5T^{2}
7 1−2.69iT−7T2 1 - 2.69iT - 7T^{2}
11 1−1.19iT−11T2 1 - 1.19iT - 11T^{2}
17 1+1.13T+17T2 1 + 1.13T + 17T^{2}
19 1−1.93iT−19T2 1 - 1.93iT - 19T^{2}
23 1+4.60T+23T2 1 + 4.60T + 23T^{2}
29 1+7.89T+29T2 1 + 7.89T + 29T^{2}
31 1−5.89iT−31T2 1 - 5.89iT - 31T^{2}
37 1−0.951iT−37T2 1 - 0.951iT - 37T^{2}
41 1+3.31iT−41T2 1 + 3.31iT - 41T^{2}
43 1−7.15T+43T2 1 - 7.15T + 43T^{2}
47 1−7.69iT−47T2 1 - 7.69iT - 47T^{2}
53 1−5.87T+53T2 1 - 5.87T + 53T^{2}
59 1−0.0120iT−59T2 1 - 0.0120iT - 59T^{2}
61 1+8.03T+61T2 1 + 8.03T + 61T^{2}
67 1+9.25iT−67T2 1 + 9.25iT - 67T^{2}
71 1+13.7iT−71T2 1 + 13.7iT - 71T^{2}
73 1−12.8iT−73T2 1 - 12.8iT - 73T^{2}
79 1+0.807T+79T2 1 + 0.807T + 79T^{2}
83 1+16.3iT−83T2 1 + 16.3iT - 83T^{2}
89 1+14.7iT−89T2 1 + 14.7iT - 89T^{2}
97 1+3.13iT−97T2 1 + 3.13iT - 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

−9.269871307609507449740371753596, −8.621897782933711997817992655171, −7.68226564481593691482252506460, −6.99164598597475051673924411488, −5.96712759408738436391867512425, −5.85541323501805077793668960039, −4.71867831282440675094683073189, −3.54225100841430409835749217595, −2.74298772545006303586897203845, −1.93042924712915159414699905062, 0.16409153340135742236739215088, 1.05592478915976210761685630646, 2.39963553934197007333555456459, 3.79483620084616979302901160719, 4.36109758089093135056752433391, 5.35792647971785701007579892626, 5.82322123255568854837122643666, 6.82294917722149886339658378389, 7.70655194517745680825838710719, 8.375834877940717778515086041323

Graph of the ZZ-function along the critical line