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Properties

Label 2-1260-5.4-c1-0-14
Degree 22
Conductor 12601260
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 10.061110.0611
Root an. cond. 3.171933.17193
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 − 2i)5-s + i·7-s − 4i·13-s − 4i·17-s − 4·19-s + 8i·23-s + (−3 + 4i)25-s + 2·29-s − 8·31-s + (2 − i)35-s − 8i·37-s − 6·41-s − 8i·43-s − 8i·47-s − 49-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + 0.377i·7-s − 1.10i·13-s − 0.970i·17-s − 0.917·19-s + 1.66i·23-s + (−0.600 + 0.800i)25-s + 0.371·29-s − 1.43·31-s + (0.338 − 0.169i)35-s − 1.31i·37-s − 0.937·41-s − 1.21i·43-s − 1.16i·47-s − 0.142·49-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 10.061110.0611
Root analytic conductor: 3.171933.17193
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1260(1009,⋅)\chi_{1260} (1009, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :1/2), −0.894+0.447i)(2,\ 1260,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.65177332360.6517733236
L(12)L(\frac12) ≈\approx 0.65177332360.6517733236
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+2i)T 1 + (1 + 2i)T
7 1−iT 1 - iT
good11 1+11T2 1 + 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1−8iT−23T2 1 - 8iT - 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1−8iT−67T2 1 - 8iT - 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+12iT−97T2 1 + 12iT - 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

−9.165611770956031616213594010138, −8.643470355764417466753512147604, −7.72514320118517489118670067005, −7.08569018465165700243735778601, −5.62085455269324287305993389993, −5.30863946361300690156724032679, −4.12284037305519620721569663355, −3.18718983780593793711955095159, −1.79452251166668040447689198556, −0.26379374275432735683046553226, 1.77944218236695912773050263148, 2.96765216725405715849112457162, 4.03685911738431397278418732169, 4.67198866454970854600655997116, 6.35548679544745067335869699082, 6.50126589139299238862603764220, 7.58758503190917037834068109774, 8.345806656197979186835781356270, 9.181292137054726356015120717343, 10.28630686549445911423429436399

Graph of the ZZ-function along the critical line