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Properties

Label 2-2640-12.11-c1-0-49
Degree 22
Conductor 26402640
Sign 0.356+0.934i0.356 + 0.934i
Analytic cond. 21.080521.0805
Root an. cond. 4.591354.59135
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  + (−1.61 + 0.618i)3-s − i·5-s + 2.07i·7-s + (2.23 − 2.00i)9-s − 11-s + 6.58·13-s + (0.618 + 1.61i)15-s − 4.07i·17-s − 4.18i·19-s + (−1.28 − 3.35i)21-s − 6.70·23-s − 25-s + (−2.38 + 4.61i)27-s + 2.18i·29-s − 8.70i·31-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)3-s − 0.447i·5-s + 0.783i·7-s + (0.745 − 0.666i)9-s − 0.301·11-s + 1.82·13-s + (0.159 + 0.417i)15-s − 0.987i·17-s − 0.960i·19-s + (−0.279 − 0.731i)21-s − 1.39·23-s − 0.200·25-s + (−0.458 + 0.888i)27-s + 0.406i·29-s − 1.56i·31-s + ⋯

Functional equation

Λ(s)=(2640s/2 ΓC(s) L(s)=((0.356+0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2640s/2 ΓC(s+1/2) L(s)=((0.356+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26402640    =    24⋅3⋅5⋅112^{4} \cdot 3 \cdot 5 \cdot 11
Sign: 0.356+0.934i0.356 + 0.934i
Analytic conductor: 21.080521.0805
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2640(1871,⋅)\chi_{2640} (1871, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2640, ( :1/2), 0.356+0.934i)(2,\ 2640,\ (\ :1/2),\ 0.356 + 0.934i)

Particular Values

L(1)L(1) ≈\approx 1.0205567151.020556715
L(12)L(\frac12) ≈\approx 1.0205567151.020556715
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+(1.61−0.618i)T 1 + (1.61 - 0.618i)T
5 1+iT 1 + iT
11 1+T 1 + T
good7 1−2.07iT−7T2 1 - 2.07iT - 7T^{2}
13 1−6.58T+13T2 1 - 6.58T + 13T^{2}
17 1+4.07iT−17T2 1 + 4.07iT - 17T^{2}
19 1+4.18iT−19T2 1 + 4.18iT - 19T^{2}
23 1+6.70T+23T2 1 + 6.70T + 23T^{2}
29 1−2.18iT−29T2 1 - 2.18iT - 29T^{2}
31 1+8.70iT−31T2 1 + 8.70iT - 31T^{2}
37 1+9.61T+37T2 1 + 9.61T + 37T^{2}
41 1−6.18iT−41T2 1 - 6.18iT - 41T^{2}
43 1−11.2iT−43T2 1 - 11.2iT - 43T^{2}
47 1−9.94T+47T2 1 - 9.94T + 47T^{2}
53 1+5.94iT−53T2 1 + 5.94iT - 53T^{2}
59 1+0.532T+59T2 1 + 0.532T + 59T^{2}
61 1−0.704T+61T2 1 - 0.704T + 61T^{2}
67 1+4.14iT−67T2 1 + 4.14iT - 67T^{2}
71 1+5.37T+71T2 1 + 5.37T + 71T^{2}
73 1+4.25T+73T2 1 + 4.25T + 73T^{2}
79 1+11.9iT−79T2 1 + 11.9iT - 79T^{2}
83 1−2.25T+83T2 1 - 2.25T + 83T^{2}
89 1+17.5iT−89T2 1 + 17.5iT - 89T^{2}
97 1−7.85T+97T2 1 - 7.85T + 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

−8.863469826626804808370613273407, −8.055826399623313865592286561723, −7.06596119284546989625097657870, −6.06661933895449422408482006575, −5.77851487622195042405480764644, −4.83377905589336601537237264774, −4.11444843170446484315048976354, −3.05856772816147809838929956894, −1.71062457221034350879448793273, −0.43937626299655193506837503797, 1.10872049940344023890306750787, 2.01574884410321890921528670043, 3.79013236488572669552551565728, 3.92185223826073096678460767909, 5.40683158417529336610661200854, 5.94837867081613155903032166984, 6.64840994858648375580463531583, 7.35996597204259991010820254591, 8.158930766430738009907060779383, 8.823902177797321474893150281094

Graph of the ZZ-function along the critical line