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Properties

Label 2-416-104.11-c1-0-10
Degree 22
Conductor 416416
Sign −0.849+0.527i-0.849 + 0.527i
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  + (−0.0597 − 0.103i)3-s + (−2.08 + 2.08i)5-s + (−1.83 + 0.491i)7-s + (1.49 − 2.58i)9-s + (−5.53 − 1.48i)11-s + (0.0282 − 3.60i)13-s + (0.340 + 0.0913i)15-s + (−3.70 − 2.13i)17-s + (4.12 − 1.10i)19-s + (0.160 + 0.160i)21-s + (−1.56 − 2.70i)23-s − 3.72i·25-s − 0.715·27-s + (−3.41 + 1.97i)29-s + (−5.91 + 5.91i)31-s + ⋯
L(s)  = 1  + (−0.0344 − 0.0597i)3-s + (−0.934 + 0.934i)5-s + (−0.693 + 0.185i)7-s + (0.497 − 0.861i)9-s + (−1.66 − 0.446i)11-s + (0.00782 − 0.999i)13-s + (0.0880 + 0.0235i)15-s + (−0.897 − 0.518i)17-s + (0.946 − 0.253i)19-s + (0.0350 + 0.0350i)21-s + (−0.325 − 0.563i)23-s − 0.745i·25-s − 0.137·27-s + (−0.634 + 0.366i)29-s + (−1.06 + 1.06i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.849+0.527i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.849+0.527i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.849+0.527i-0.849 + 0.527i
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), −0.849+0.527i)(2,\ 416,\ (\ :1/2),\ -0.849 + 0.527i)

Particular Values

L(1)L(1) ≈\approx 0.0610533−0.213996i0.0610533 - 0.213996i
L(12)L(\frac12) ≈\approx 0.0610533−0.213996i0.0610533 - 0.213996i
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+(−0.0282+3.60i)T 1 + (-0.0282 + 3.60i)T
good3 1+(0.0597+0.103i)T+(−1.5+2.59i)T2 1 + (0.0597 + 0.103i)T + (-1.5 + 2.59i)T^{2}
5 1+(2.08−2.08i)T−5iT2 1 + (2.08 - 2.08i)T - 5iT^{2}
7 1+(1.83−0.491i)T+(6.06−3.5i)T2 1 + (1.83 - 0.491i)T + (6.06 - 3.5i)T^{2}
11 1+(5.53+1.48i)T+(9.52+5.5i)T2 1 + (5.53 + 1.48i)T + (9.52 + 5.5i)T^{2}
17 1+(3.70+2.13i)T+(8.5+14.7i)T2 1 + (3.70 + 2.13i)T + (8.5 + 14.7i)T^{2}
19 1+(−4.12+1.10i)T+(16.4−9.5i)T2 1 + (-4.12 + 1.10i)T + (16.4 - 9.5i)T^{2}
23 1+(1.56+2.70i)T+(−11.5+19.9i)T2 1 + (1.56 + 2.70i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.41−1.97i)T+(14.5−25.1i)T2 1 + (3.41 - 1.97i)T + (14.5 - 25.1i)T^{2}
31 1+(5.91−5.91i)T−31iT2 1 + (5.91 - 5.91i)T - 31iT^{2}
37 1+(0.0218−0.0814i)T+(−32.0−18.5i)T2 1 + (0.0218 - 0.0814i)T + (-32.0 - 18.5i)T^{2}
41 1+(1.89−7.07i)T+(−35.5−20.5i)T2 1 + (1.89 - 7.07i)T + (-35.5 - 20.5i)T^{2}
43 1+(−3.96−2.28i)T+(21.5+37.2i)T2 1 + (-3.96 - 2.28i)T + (21.5 + 37.2i)T^{2}
47 1+(−1.33−1.33i)T+47iT2 1 + (-1.33 - 1.33i)T + 47iT^{2}
53 1+7.65iT−53T2 1 + 7.65iT - 53T^{2}
59 1+(0.332+1.23i)T+(−51.0+29.5i)T2 1 + (0.332 + 1.23i)T + (-51.0 + 29.5i)T^{2}
61 1+(5.12+2.95i)T+(30.5+52.8i)T2 1 + (5.12 + 2.95i)T + (30.5 + 52.8i)T^{2}
67 1+(0.943−3.52i)T+(−58.0−33.5i)T2 1 + (0.943 - 3.52i)T + (-58.0 - 33.5i)T^{2}
71 1+(1.87+6.98i)T+(−61.4+35.5i)T2 1 + (1.87 + 6.98i)T + (-61.4 + 35.5i)T^{2}
73 1+(2.35−2.35i)T−73iT2 1 + (2.35 - 2.35i)T - 73iT^{2}
79 1+4.48iT−79T2 1 + 4.48iT - 79T^{2}
83 1+(0.871+0.871i)T+83iT2 1 + (0.871 + 0.871i)T + 83iT^{2}
89 1+(0.761+0.204i)T+(77.0+44.5i)T2 1 + (0.761 + 0.204i)T + (77.0 + 44.5i)T^{2}
97 1+(−11.8+3.18i)T+(84.0−48.5i)T2 1 + (-11.8 + 3.18i)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

−10.83136610905134386353918849243, −10.11829826503382618448858807804, −9.043542094314991423204339600828, −7.83401023351452000154507038042, −7.20951971853189919703015286500, −6.22099859675046300523166166618, −5.01587347882311075681856569336, −3.45170685483473557780746556358, −2.87994524550062699555559670175, −0.13530578043297229424211231585, 2.09967550700940040725919181612, 3.84973400691633183750223291300, 4.68238085309089028128228035497, 5.68465384852507084837009857144, 7.31788577128033686607840914799, 7.72109871794912207651550986995, 8.839591858113367345709770906727, 9.783292211111754404344773578959, 10.69036758427617993263424511740, 11.58884375772164680767089512166

Graph of the ZZ-function along the critical line