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Properties

Label 2-416-104.19-c1-0-7
Degree 22
Conductor 416416
Sign 0.972+0.231i0.972 + 0.231i
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  + (0.223 − 0.386i)3-s + (−0.612 − 0.612i)5-s + (2.00 + 0.537i)7-s + (1.40 + 2.42i)9-s + (3.04 − 0.816i)11-s + (−3.45 − 1.02i)13-s + (−0.373 + 0.100i)15-s + (2.68 − 1.55i)17-s + (3.53 + 0.948i)19-s + (0.655 − 0.655i)21-s + (1.56 − 2.70i)23-s − 4.24i·25-s + 2.58·27-s + (6.13 + 3.54i)29-s + (−2.77 − 2.77i)31-s + ⋯
L(s)  = 1  + (0.128 − 0.223i)3-s + (−0.274 − 0.274i)5-s + (0.758 + 0.203i)7-s + (0.466 + 0.808i)9-s + (0.918 − 0.246i)11-s + (−0.958 − 0.283i)13-s + (−0.0964 + 0.0258i)15-s + (0.651 − 0.376i)17-s + (0.812 + 0.217i)19-s + (0.143 − 0.143i)21-s + (0.325 − 0.563i)23-s − 0.849i·25-s + 0.497·27-s + (1.13 + 0.657i)29-s + (−0.498 − 0.498i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.972+0.231i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.972+0.231i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.972+0.231i0.972 + 0.231i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(175,⋅)\chi_{416} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.972+0.231i)(2,\ 416,\ (\ :1/2),\ 0.972 + 0.231i)

Particular Values

L(1)L(1) ≈\approx 1.56515−0.183987i1.56515 - 0.183987i
L(12)L(\frac12) ≈\approx 1.56515−0.183987i1.56515 - 0.183987i
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+(3.45+1.02i)T 1 + (3.45 + 1.02i)T
good3 1+(−0.223+0.386i)T+(−1.5−2.59i)T2 1 + (-0.223 + 0.386i)T + (-1.5 - 2.59i)T^{2}
5 1+(0.612+0.612i)T+5iT2 1 + (0.612 + 0.612i)T + 5iT^{2}
7 1+(−2.00−0.537i)T+(6.06+3.5i)T2 1 + (-2.00 - 0.537i)T + (6.06 + 3.5i)T^{2}
11 1+(−3.04+0.816i)T+(9.52−5.5i)T2 1 + (-3.04 + 0.816i)T + (9.52 - 5.5i)T^{2}
17 1+(−2.68+1.55i)T+(8.5−14.7i)T2 1 + (-2.68 + 1.55i)T + (8.5 - 14.7i)T^{2}
19 1+(−3.53−0.948i)T+(16.4+9.5i)T2 1 + (-3.53 - 0.948i)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+(−6.13−3.54i)T+(14.5+25.1i)T2 1 + (-6.13 - 3.54i)T + (14.5 + 25.1i)T^{2}
31 1+(2.77+2.77i)T+31iT2 1 + (2.77 + 2.77i)T + 31iT^{2}
37 1+(0.155+0.580i)T+(−32.0+18.5i)T2 1 + (0.155 + 0.580i)T + (-32.0 + 18.5i)T^{2}
41 1+(−1.60−5.98i)T+(−35.5+20.5i)T2 1 + (-1.60 - 5.98i)T + (-35.5 + 20.5i)T^{2}
43 1+(5.34−3.08i)T+(21.5−37.2i)T2 1 + (5.34 - 3.08i)T + (21.5 - 37.2i)T^{2}
47 1+(6.19−6.19i)T−47iT2 1 + (6.19 - 6.19i)T - 47iT^{2}
53 1−2.19iT−53T2 1 - 2.19iT - 53T^{2}
59 1+(1.67−6.25i)T+(−51.0−29.5i)T2 1 + (1.67 - 6.25i)T + (-51.0 - 29.5i)T^{2}
61 1+(−0.346+0.199i)T+(30.5−52.8i)T2 1 + (-0.346 + 0.199i)T + (30.5 - 52.8i)T^{2}
67 1+(2.40+8.98i)T+(−58.0+33.5i)T2 1 + (2.40 + 8.98i)T + (-58.0 + 33.5i)T^{2}
71 1+(−4.33+16.1i)T+(−61.4−35.5i)T2 1 + (-4.33 + 16.1i)T + (-61.4 - 35.5i)T^{2}
73 1+(6.53+6.53i)T+73iT2 1 + (6.53 + 6.53i)T + 73iT^{2}
79 1−12.9iT−79T2 1 - 12.9iT - 79T^{2}
83 1+(9.22−9.22i)T−83iT2 1 + (9.22 - 9.22i)T - 83iT^{2}
89 1+(6.08−1.62i)T+(77.0−44.5i)T2 1 + (6.08 - 1.62i)T + (77.0 - 44.5i)T^{2}
97 1+(15.1+4.07i)T+(84.0+48.5i)T2 1 + (15.1 + 4.07i)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

−11.29871843418456045845888548079, −10.26004541864723758229179770849, −9.364110040422578806627685786653, −8.212416966297327009278373164039, −7.68508560775830851141880177520, −6.58519463675182895629420143452, −5.17426342275804927652395076663, −4.48803804198555070520764574759, −2.88470963708672234920547046008, −1.37344641690306194952559351537, 1.45007319665933950138927253952, 3.28192474755651532634963529183, 4.27179277743187263515410008059, 5.34388766243379653224521765932, 6.81080893520165300282656097253, 7.38148179989101068316719898747, 8.564339647665530194706464304380, 9.557248352843997550173033090843, 10.17049612628388713418499636237, 11.47918290541830145841476747090

Graph of the ZZ-function along the critical line