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Properties

Label 2-4680-5.4-c1-0-81
Degree 22
Conductor 46804680
Sign −0.783+0.621i-0.783 + 0.621i
Analytic cond. 37.369937.3699
Root an. cond. 6.113096.11309
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  + (1.75 − 1.38i)5-s − 1.50·11-s − i·13-s + 2.72i·17-s − 0.726·19-s − 4.72i·23-s + (1.14 − 4.86i)25-s − 7.55·29-s − 3.00·31-s − 5.00i·37-s − 5.78·41-s − 2.72i·43-s + 10.2i·47-s + 7·49-s − 7.55i·53-s + ⋯
L(s)  = 1  + (0.783 − 0.621i)5-s − 0.453·11-s − 0.277i·13-s + 0.661i·17-s − 0.166·19-s − 0.985i·23-s + (0.228 − 0.973i)25-s − 1.40·29-s − 0.540·31-s − 0.823i·37-s − 0.903·41-s − 0.415i·43-s + 1.49i·47-s + 49-s − 1.03i·53-s + ⋯

Functional equation

Λ(s)=(4680s/2 ΓC(s) L(s)=((−0.783+0.621i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4680s/2 ΓC(s+1/2) L(s)=((−0.783+0.621i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46804680    =    23⋅32⋅5⋅132^{3} \cdot 3^{2} \cdot 5 \cdot 13
Sign: −0.783+0.621i-0.783 + 0.621i
Analytic conductor: 37.369937.3699
Root analytic conductor: 6.113096.11309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4680(2809,⋅)\chi_{4680} (2809, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4680, ( :1/2), −0.783+0.621i)(2,\ 4680,\ (\ :1/2),\ -0.783 + 0.621i)

Particular Values

L(1)L(1) ≈\approx 1.0981593971.098159397
L(12)L(\frac12) ≈\approx 1.0981593971.098159397
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+(−1.75+1.38i)T 1 + (-1.75 + 1.38i)T
13 1+iT 1 + iT
good7 1−7T2 1 - 7T^{2}
11 1+1.50T+11T2 1 + 1.50T + 11T^{2}
17 1−2.72iT−17T2 1 - 2.72iT - 17T^{2}
19 1+0.726T+19T2 1 + 0.726T + 19T^{2}
23 1+4.72iT−23T2 1 + 4.72iT - 23T^{2}
29 1+7.55T+29T2 1 + 7.55T + 29T^{2}
31 1+3.00T+31T2 1 + 3.00T + 31T^{2}
37 1+5.00iT−37T2 1 + 5.00iT - 37T^{2}
41 1+5.78T+41T2 1 + 5.78T + 41T^{2}
43 1+2.72iT−43T2 1 + 2.72iT - 43T^{2}
47 1−10.2iT−47T2 1 - 10.2iT - 47T^{2}
53 1+7.55iT−53T2 1 + 7.55iT - 53T^{2}
59 1+12.5T+59T2 1 + 12.5T + 59T^{2}
61 1−6.28T+61T2 1 - 6.28T + 61T^{2}
67 1+12.5iT−67T2 1 + 12.5iT - 67T^{2}
71 1+4.77T+71T2 1 + 4.77T + 71T^{2}
73 1+12.0iT−73T2 1 + 12.0iT - 73T^{2}
79 1+5.27T+79T2 1 + 5.27T + 79T^{2}
83 1−7.78iT−83T2 1 - 7.78iT - 83T^{2}
89 1−1.78T+89T2 1 - 1.78T + 89T^{2}
97 1+6iT−97T2 1 + 6iT - 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

−8.067280939904542458198515079497, −7.38362675912891075622090017153, −6.39946989738521429270424756177, −5.81549909634357433015035795755, −5.14038688791482945576165117922, −4.37472819897967762719263278948, −3.43492733731255321985367912248, −2.34458374553479956325734422953, −1.61642080019442197313070869995, −0.27553392848357052012657172700, 1.43977903962926244005598572791, 2.32376218571008333153546126415, 3.14935217996270840335666641041, 3.99632613537478789566895332304, 5.14440111942479451691307537530, 5.60021254918280908032851939676, 6.42591721172297187669787477953, 7.21348383142416693559525341412, 7.62723007256731898107850275040, 8.759461530740780467180565002304

Graph of the ZZ-function along the critical line