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Properties

Label 2-3872-8.5-c1-0-78
Degree 22
Conductor 38723872
Sign −0.229+0.973i-0.229 + 0.973i
Analytic cond. 30.918030.9180
Root an. cond. 5.560405.56040
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  − 0.540i·3-s − 2.90i·5-s + 2.60·7-s + 2.70·9-s + 2.61i·13-s − 1.56·15-s − 6.89·17-s − 1.60i·19-s − 1.40i·21-s + 4.01·23-s − 3.41·25-s − 3.08i·27-s − 1.99i·29-s + 5.18·31-s − 7.55i·35-s + ⋯
L(s)  = 1  − 0.312i·3-s − 1.29i·5-s + 0.984·7-s + 0.902·9-s + 0.725i·13-s − 0.404·15-s − 1.67·17-s − 0.367i·19-s − 0.307i·21-s + 0.837·23-s − 0.683·25-s − 0.593i·27-s − 0.370i·29-s + 0.930·31-s − 1.27i·35-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((−0.229+0.973i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s+1/2) L(s)=((−0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: −0.229+0.973i-0.229 + 0.973i
Analytic conductor: 30.918030.9180
Root analytic conductor: 5.560405.56040
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3872(1937,⋅)\chi_{3872} (1937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :1/2), −0.229+0.973i)(2,\ 3872,\ (\ :1/2),\ -0.229 + 0.973i)

Particular Values

L(1)L(1) ≈\approx 2.1450189132.145018913
L(12)L(\frac12) ≈\approx 2.1450189132.145018913
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
11 1 1
good3 1+0.540iT−3T2 1 + 0.540iT - 3T^{2}
5 1+2.90iT−5T2 1 + 2.90iT - 5T^{2}
7 1−2.60T+7T2 1 - 2.60T + 7T^{2}
13 1−2.61iT−13T2 1 - 2.61iT - 13T^{2}
17 1+6.89T+17T2 1 + 6.89T + 17T^{2}
19 1+1.60iT−19T2 1 + 1.60iT - 19T^{2}
23 1−4.01T+23T2 1 - 4.01T + 23T^{2}
29 1+1.99iT−29T2 1 + 1.99iT - 29T^{2}
31 1−5.18T+31T2 1 - 5.18T + 31T^{2}
37 1+8.58iT−37T2 1 + 8.58iT - 37T^{2}
41 1+3.74T+41T2 1 + 3.74T + 41T^{2}
43 1+2.47iT−43T2 1 + 2.47iT - 43T^{2}
47 1−11.8T+47T2 1 - 11.8T + 47T^{2}
53 1−1.29iT−53T2 1 - 1.29iT - 53T^{2}
59 1+0.351iT−59T2 1 + 0.351iT - 59T^{2}
61 1+3.08iT−61T2 1 + 3.08iT - 61T^{2}
67 1+10.3iT−67T2 1 + 10.3iT - 67T^{2}
71 1+3.47T+71T2 1 + 3.47T + 71T^{2}
73 1+1.33T+73T2 1 + 1.33T + 73T^{2}
79 1−4.32T+79T2 1 - 4.32T + 79T^{2}
83 1+15.9iT−83T2 1 + 15.9iT - 83T^{2}
89 1+13.1T+89T2 1 + 13.1T + 89T^{2}
97 1−8.61T+97T2 1 - 8.61T + 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

−8.393568992762305858202574827940, −7.52851104089077145076808019042, −6.92584122312136918257336685748, −6.06981774831427870258557617624, −4.91996480219444093552520603173, −4.64545951421078829128223774765, −3.97151008553635904341389479707, −2.32164494305229086493011179422, −1.63545306923616809526338703094, −0.66334738858411860620795175900, 1.28743204027936239305134018848, 2.39716711639003532257263345048, 3.15956100474084994765569842297, 4.21528549168006274628991489856, 4.76842825168096123534822413027, 5.71258458071706774213012651753, 6.83685769239310709693162788545, 6.95120450150727007565184786172, 7.975354137257398452244679188064, 8.568007250756743754863884383744

Graph of the ZZ-function along the critical line