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Properties

Label 2-40e2-80.27-c1-0-32
Degree 22
Conductor 16001600
Sign −0.584−0.811i-0.584 - 0.811i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + (−3 − 3i)7-s + 9-s + (1 − i)11-s − 2i·13-s + (−1 − i)17-s + (3 − 3i)19-s + (6 + 6i)21-s + (−1 + i)23-s + 4·27-s + (−7 − 7i)29-s + 2i·31-s + (−2 + 2i)33-s + 6i·37-s + 4i·39-s + ⋯
L(s)  = 1  − 1.15·3-s + (−1.13 − 1.13i)7-s + 0.333·9-s + (0.301 − 0.301i)11-s − 0.554i·13-s + (−0.242 − 0.242i)17-s + (0.688 − 0.688i)19-s + (1.30 + 1.30i)21-s + (−0.208 + 0.208i)23-s + 0.769·27-s + (−1.29 − 1.29i)29-s + 0.359i·31-s + (−0.348 + 0.348i)33-s + 0.986i·37-s + 0.640i·39-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.584−0.811i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((−0.584−0.811i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.584−0.811i-0.584 - 0.811i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(207,⋅)\chi_{1600} (207, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1600, ( :1/2), −0.584−0.811i)(2,\ 1600,\ (\ :1/2),\ -0.584 - 0.811i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1 1
good3 1+2T+3T2 1 + 2T + 3T^{2}
7 1+(3+3i)T+7iT2 1 + (3 + 3i)T + 7iT^{2}
11 1+(−1+i)T−11iT2 1 + (-1 + i)T - 11iT^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1+(1+i)T+17iT2 1 + (1 + i)T + 17iT^{2}
19 1+(−3+3i)T−19iT2 1 + (-3 + 3i)T - 19iT^{2}
23 1+(1−i)T−23iT2 1 + (1 - i)T - 23iT^{2}
29 1+(7+7i)T+29iT2 1 + (7 + 7i)T + 29iT^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1−4iT−41T2 1 - 4iT - 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+(7−7i)T−47iT2 1 + (7 - 7i)T - 47iT^{2}
53 1−8T+53T2 1 - 8T + 53T^{2}
59 1+(3+3i)T+59iT2 1 + (3 + 3i)T + 59iT^{2}
61 1+(1−i)T−61iT2 1 + (1 - i)T - 61iT^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+(3+3i)T+73iT2 1 + (3 + 3i)T + 73iT^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+2T+83T2 1 + 2T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(−11−11i)T+97iT2 1 + (-11 - 11i)T + 97iT^{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

−9.101139535687888154757343634177, −7.896187354212271565040723769402, −7.07867655731667953341167169694, −6.41390294988502804524204609402, −5.74396736749512175495356481205, −4.80692827445366092158236224068, −3.81070009470153480989315449889, −2.90384284904392945037127602770, −0.992545958293620806338961011892, 0, 1.80318515915811479115274602692, 3.07283504649750636870310439265, 4.11119843435503972832553631872, 5.37856479852763345216332433780, 5.75715420349267141210556490114, 6.56719111629516412301280577319, 7.22255523085825466519027665419, 8.532386326098033075536076699371, 9.238942270210354711088330763436

Graph of the ZZ-function along the critical line