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Properties

Label 2-78e2-13.12-c1-0-26
Degree 22
Conductor 60846084
Sign 0.832−0.554i0.832 - 0.554i
Analytic cond. 48.580948.5809
Root an. cond. 6.970006.97000
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  + 2.64i·5-s + 2i·7-s − 5.29i·11-s − 7.93·17-s − 6i·19-s + 5.29·23-s − 2.00·25-s + 2.64·29-s + 4i·31-s − 5.29·35-s + 3i·37-s + 7.93i·41-s + 2·43-s − 5.29i·47-s + 3·49-s + ⋯
L(s)  = 1  + 1.18i·5-s + 0.755i·7-s − 1.59i·11-s − 1.92·17-s − 1.37i·19-s + 1.10·23-s − 0.400·25-s + 0.491·29-s + 0.718i·31-s − 0.894·35-s + 0.493i·37-s + 1.23i·41-s + 0.304·43-s − 0.771i·47-s + 0.428·49-s + ⋯

Functional equation

Λ(s)=(6084s/2 ΓC(s) L(s)=((0.832−0.554i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6084s/2 ΓC(s+1/2) L(s)=((0.832−0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60846084    =    22⋅32⋅1322^{2} \cdot 3^{2} \cdot 13^{2}
Sign: 0.832−0.554i0.832 - 0.554i
Analytic conductor: 48.580948.5809
Root analytic conductor: 6.970006.97000
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6084(4393,⋅)\chi_{6084} (4393, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6084, ( :1/2), 0.832−0.554i)(2,\ 6084,\ (\ :1/2),\ 0.832 - 0.554i)

Particular Values

L(1)L(1) ≈\approx 1.7603906101.760390610
L(12)L(\frac12) ≈\approx 1.7603906101.760390610
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
13 1 1
good5 1−2.64iT−5T2 1 - 2.64iT - 5T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+5.29iT−11T2 1 + 5.29iT - 11T^{2}
17 1+7.93T+17T2 1 + 7.93T + 17T^{2}
19 1+6iT−19T2 1 + 6iT - 19T^{2}
23 1−5.29T+23T2 1 - 5.29T + 23T^{2}
29 1−2.64T+29T2 1 - 2.64T + 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1−3iT−37T2 1 - 3iT - 37T^{2}
41 1−7.93iT−41T2 1 - 7.93iT - 41T^{2}
43 1−2T+43T2 1 - 2T + 43T^{2}
47 1+5.29iT−47T2 1 + 5.29iT - 47T^{2}
53 1−7.93T+53T2 1 - 7.93T + 53T^{2}
59 1+10.5iT−59T2 1 + 10.5iT - 59T^{2}
61 1−13T+61T2 1 - 13T + 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1−5.29iT−71T2 1 - 5.29iT - 71T^{2}
73 1−7iT−73T2 1 - 7iT - 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+15.8iT−83T2 1 + 15.8iT - 83T^{2}
89 1−89T2 1 - 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

−8.419188823677209357125077658079, −7.14117125961160430318399680623, −6.72194648808355840947704634348, −6.20527302271423644290530842961, −5.30743652657570578768211416349, −4.58146860233411240892781943507, −3.44553340141929751749511229439, −2.82081095737370415561386707199, −2.28995213211773416978028764483, −0.70145465678015439861907802134, 0.68224575516304149380698444799, 1.70087814950988463366599004605, 2.47414777257690956838102414481, 4.00881949301678191272096330797, 4.27374362121762830318721838309, 4.98082451996679712795097714269, 5.75745398886677038213694393374, 6.81738181849805058682072567329, 7.21582222640926262336251573311, 8.001422117862306524346951103087

Graph of the ZZ-function along the critical line