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Properties

Label 2-416-104.11-c1-0-11
Degree 22
Conductor 416416
Sign −0.735−0.677i-0.735 - 0.677i
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  + (−1.11 − 1.92i)3-s + (0.0693 − 0.0693i)5-s + (−3.36 + 0.901i)7-s + (−0.979 + 1.69i)9-s + (0.305 + 0.0819i)11-s + (−3.13 + 1.78i)13-s + (−0.210 − 0.0565i)15-s + (−5.48 − 3.16i)17-s + (0.397 − 0.106i)19-s + (5.48 + 5.48i)21-s + (3.68 + 6.38i)23-s + 4.99i·25-s − 2.32·27-s + (−3.16 + 1.82i)29-s + (−1.12 + 1.12i)31-s + ⋯
L(s)  = 1  + (−0.642 − 1.11i)3-s + (0.0310 − 0.0310i)5-s + (−1.27 + 0.340i)7-s + (−0.326 + 0.565i)9-s + (0.0921 + 0.0247i)11-s + (−0.868 + 0.495i)13-s + (−0.0544 − 0.0145i)15-s + (−1.33 − 0.768i)17-s + (0.0911 − 0.0244i)19-s + (1.19 + 1.19i)21-s + (0.768 + 1.33i)23-s + 0.998i·25-s − 0.446·27-s + (−0.588 + 0.339i)29-s + (−0.201 + 0.201i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.735−0.677i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.735−0.677i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.735−0.677i-0.735 - 0.677i
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), −0.735−0.677i)(2,\ 416,\ (\ :1/2),\ -0.735 - 0.677i)

Particular Values

L(1)L(1) ≈\approx 0.0166899+0.0427574i0.0166899 + 0.0427574i
L(12)L(\frac12) ≈\approx 0.0166899+0.0427574i0.0166899 + 0.0427574i
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+(3.13−1.78i)T 1 + (3.13 - 1.78i)T
good3 1+(1.11+1.92i)T+(−1.5+2.59i)T2 1 + (1.11 + 1.92i)T + (-1.5 + 2.59i)T^{2}
5 1+(−0.0693+0.0693i)T−5iT2 1 + (-0.0693 + 0.0693i)T - 5iT^{2}
7 1+(3.36−0.901i)T+(6.06−3.5i)T2 1 + (3.36 - 0.901i)T + (6.06 - 3.5i)T^{2}
11 1+(−0.305−0.0819i)T+(9.52+5.5i)T2 1 + (-0.305 - 0.0819i)T + (9.52 + 5.5i)T^{2}
17 1+(5.48+3.16i)T+(8.5+14.7i)T2 1 + (5.48 + 3.16i)T + (8.5 + 14.7i)T^{2}
19 1+(−0.397+0.106i)T+(16.4−9.5i)T2 1 + (-0.397 + 0.106i)T + (16.4 - 9.5i)T^{2}
23 1+(−3.68−6.38i)T+(−11.5+19.9i)T2 1 + (-3.68 - 6.38i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.16−1.82i)T+(14.5−25.1i)T2 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2}
31 1+(1.12−1.12i)T−31iT2 1 + (1.12 - 1.12i)T - 31iT^{2}
37 1+(−0.804+3.00i)T+(−32.0−18.5i)T2 1 + (-0.804 + 3.00i)T + (-32.0 - 18.5i)T^{2}
41 1+(−1.85+6.91i)T+(−35.5−20.5i)T2 1 + (-1.85 + 6.91i)T + (-35.5 - 20.5i)T^{2}
43 1+(4.88+2.82i)T+(21.5+37.2i)T2 1 + (4.88 + 2.82i)T + (21.5 + 37.2i)T^{2}
47 1+(6.15+6.15i)T+47iT2 1 + (6.15 + 6.15i)T + 47iT^{2}
53 1−3.89iT−53T2 1 - 3.89iT - 53T^{2}
59 1+(−2.51−9.38i)T+(−51.0+29.5i)T2 1 + (-2.51 - 9.38i)T + (-51.0 + 29.5i)T^{2}
61 1+(11.9+6.88i)T+(30.5+52.8i)T2 1 + (11.9 + 6.88i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.16+11.8i)T+(−58.0−33.5i)T2 1 + (-3.16 + 11.8i)T + (-58.0 - 33.5i)T^{2}
71 1+(1.81+6.78i)T+(−61.4+35.5i)T2 1 + (1.81 + 6.78i)T + (-61.4 + 35.5i)T^{2}
73 1+(−1.49+1.49i)T−73iT2 1 + (-1.49 + 1.49i)T - 73iT^{2}
79 1+10.3iT−79T2 1 + 10.3iT - 79T^{2}
83 1+(2.53+2.53i)T+83iT2 1 + (2.53 + 2.53i)T + 83iT^{2}
89 1+(−0.110−0.0296i)T+(77.0+44.5i)T2 1 + (-0.110 - 0.0296i)T + (77.0 + 44.5i)T^{2}
97 1+(−6.28+1.68i)T+(84.0−48.5i)T2 1 + (-6.28 + 1.68i)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

−10.90413344058827020232826635867, −9.471131725698682293257869361388, −9.120157491401255198346623103186, −7.34575926149391197019399540321, −7.00261361513033637645140801573, −6.06065688122737281705935328249, −5.04854973183695569466040269709, −3.36570430643453266443203606513, −1.94131753314559723368017901025, −0.02979527907365737236145918820, 2.76511263215462049340564738413, 4.07319329257073920655319450545, 4.85056821280595744400649037978, 6.12348335037236385888045537659, 6.81535522783244125645748530173, 8.244653512599268273693937680078, 9.435014892018870551809125194259, 10.02014088901205016153959024391, 10.69257906171311018407543072522, 11.51523840311068410902391198781

Graph of the ZZ-function along the critical line