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Properties

Label 2-416-104.11-c1-0-7
Degree 22
Conductor 416416
Sign 0.975−0.219i0.975 - 0.219i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  + (0.928 + 1.60i)3-s + (1.51 − 1.51i)5-s + (2.97 − 0.797i)7-s + (−0.223 + 0.386i)9-s + (−0.865 − 0.231i)11-s + (−0.159 − 3.60i)13-s + (3.84 + 1.03i)15-s + (1.05 + 0.610i)17-s + (−6.68 + 1.79i)19-s + (4.04 + 4.04i)21-s + (−0.433 − 0.751i)23-s + 0.390i·25-s + 4.74·27-s + (3.26 − 1.88i)29-s + (−5.06 + 5.06i)31-s + ⋯
L(s)  = 1  + (0.535 + 0.928i)3-s + (0.678 − 0.678i)5-s + (1.12 − 0.301i)7-s + (−0.0743 + 0.128i)9-s + (−0.260 − 0.0699i)11-s + (−0.0442 − 0.999i)13-s + (0.994 + 0.266i)15-s + (0.256 + 0.147i)17-s + (−1.53 + 0.410i)19-s + (0.883 + 0.883i)21-s + (−0.0904 − 0.156i)23-s + 0.0780i·25-s + 0.912·27-s + (0.606 − 0.350i)29-s + (−0.909 + 0.909i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.975−0.219i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.975−0.219i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.975−0.219i0.975 - 0.219i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(271,⋅)\chi_{416} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.975−0.219i)(2,\ 416,\ (\ :1/2),\ 0.975 - 0.219i)

Particular Values

L(1)L(1) ≈\approx 1.94492+0.216075i1.94492 + 0.216075i
L(12)L(\frac12) ≈\approx 1.94492+0.216075i1.94492 + 0.216075i
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
13 1+(0.159+3.60i)T 1 + (0.159 + 3.60i)T
good3 1+(−0.928−1.60i)T+(−1.5+2.59i)T2 1 + (-0.928 - 1.60i)T + (-1.5 + 2.59i)T^{2}
5 1+(−1.51+1.51i)T−5iT2 1 + (-1.51 + 1.51i)T - 5iT^{2}
7 1+(−2.97+0.797i)T+(6.06−3.5i)T2 1 + (-2.97 + 0.797i)T + (6.06 - 3.5i)T^{2}
11 1+(0.865+0.231i)T+(9.52+5.5i)T2 1 + (0.865 + 0.231i)T + (9.52 + 5.5i)T^{2}
17 1+(−1.05−0.610i)T+(8.5+14.7i)T2 1 + (-1.05 - 0.610i)T + (8.5 + 14.7i)T^{2}
19 1+(6.68−1.79i)T+(16.4−9.5i)T2 1 + (6.68 - 1.79i)T + (16.4 - 9.5i)T^{2}
23 1+(0.433+0.751i)T+(−11.5+19.9i)T2 1 + (0.433 + 0.751i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.26+1.88i)T+(14.5−25.1i)T2 1 + (-3.26 + 1.88i)T + (14.5 - 25.1i)T^{2}
31 1+(5.06−5.06i)T−31iT2 1 + (5.06 - 5.06i)T - 31iT^{2}
37 1+(2.52−9.43i)T+(−32.0−18.5i)T2 1 + (2.52 - 9.43i)T + (-32.0 - 18.5i)T^{2}
41 1+(3.12−11.6i)T+(−35.5−20.5i)T2 1 + (3.12 - 11.6i)T + (-35.5 - 20.5i)T^{2}
43 1+(−4.22−2.44i)T+(21.5+37.2i)T2 1 + (-4.22 - 2.44i)T + (21.5 + 37.2i)T^{2}
47 1+(4.24+4.24i)T+47iT2 1 + (4.24 + 4.24i)T + 47iT^{2}
53 1+2.16iT−53T2 1 + 2.16iT - 53T^{2}
59 1+(0.0382+0.142i)T+(−51.0+29.5i)T2 1 + (0.0382 + 0.142i)T + (-51.0 + 29.5i)T^{2}
61 1+(7.26+4.19i)T+(30.5+52.8i)T2 1 + (7.26 + 4.19i)T + (30.5 + 52.8i)T^{2}
67 1+(0.422−1.57i)T+(−58.0−33.5i)T2 1 + (0.422 - 1.57i)T + (-58.0 - 33.5i)T^{2}
71 1+(4.27+15.9i)T+(−61.4+35.5i)T2 1 + (4.27 + 15.9i)T + (-61.4 + 35.5i)T^{2}
73 1+(−9.55+9.55i)T−73iT2 1 + (-9.55 + 9.55i)T - 73iT^{2}
79 1−6.37iT−79T2 1 - 6.37iT - 79T^{2}
83 1+(3.53+3.53i)T+83iT2 1 + (3.53 + 3.53i)T + 83iT^{2}
89 1+(−9.33−2.50i)T+(77.0+44.5i)T2 1 + (-9.33 - 2.50i)T + (77.0 + 44.5i)T^{2}
97 1+(10.6−2.86i)T+(84.0−48.5i)T2 1 + (10.6 - 2.86i)T + (84.0 - 48.5i)T^{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

−10.88412257036020915924711732649, −10.31416530393535063949346672193, −9.484184866149994461080074400154, −8.461911619318723675417298077120, −8.017481286586547638788382159051, −6.38715453477412285320767931102, −5.12530979972171193359674368991, −4.52805714218046460337257854617, −3.21209303734983575832333975621, −1.57735721613116801150217085813, 1.88049241371162850629691024157, 2.40752838336305990880120502049, 4.27031566540695760861872321502, 5.55411239033599254338840227950, 6.69851526660139371633523873470, 7.40622444225036150606662708188, 8.388265408137165573412509129714, 9.141676156057092273566901358835, 10.46085532704549707229054679690, 11.08787039837525676122349875191

Graph of the ZZ-function along the critical line