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Properties

Label 2-416-104.19-c1-0-6
Degree 22
Conductor 416416
Sign 0.894+0.446i0.894 + 0.446i
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  + (−1.05 + 1.82i)3-s + (−2.29 − 2.29i)5-s + (−1.21 − 0.324i)7-s + (−0.722 − 1.25i)9-s + (3.76 − 1.00i)11-s + (3.23 − 1.59i)13-s + (6.59 − 1.76i)15-s + (3.99 − 2.30i)17-s + (−2.52 − 0.675i)19-s + (1.86 − 1.86i)21-s + (1.62 − 2.82i)23-s + 5.49i·25-s − 3.27·27-s + (7.35 + 4.24i)29-s + (−2.29 − 2.29i)31-s + ⋯
L(s)  = 1  + (−0.608 + 1.05i)3-s + (−1.02 − 1.02i)5-s + (−0.457 − 0.122i)7-s + (−0.240 − 0.417i)9-s + (1.13 − 0.304i)11-s + (0.896 − 0.442i)13-s + (1.70 − 0.456i)15-s + (0.969 − 0.559i)17-s + (−0.578 − 0.154i)19-s + (0.407 − 0.407i)21-s + (0.339 − 0.588i)23-s + 1.09i·25-s − 0.630·27-s + (1.36 + 0.788i)29-s + (−0.411 − 0.411i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.894+0.446i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.894+0.446i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.894+0.446i0.894 + 0.446i
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), 0.894+0.446i)(2,\ 416,\ (\ :1/2),\ 0.894 + 0.446i)

Particular Values

L(1)L(1) ≈\approx 0.881162−0.207430i0.881162 - 0.207430i
L(12)L(\frac12) ≈\approx 0.881162−0.207430i0.881162 - 0.207430i
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+(−3.23+1.59i)T 1 + (-3.23 + 1.59i)T
good3 1+(1.05−1.82i)T+(−1.5−2.59i)T2 1 + (1.05 - 1.82i)T + (-1.5 - 2.59i)T^{2}
5 1+(2.29+2.29i)T+5iT2 1 + (2.29 + 2.29i)T + 5iT^{2}
7 1+(1.21+0.324i)T+(6.06+3.5i)T2 1 + (1.21 + 0.324i)T + (6.06 + 3.5i)T^{2}
11 1+(−3.76+1.00i)T+(9.52−5.5i)T2 1 + (-3.76 + 1.00i)T + (9.52 - 5.5i)T^{2}
17 1+(−3.99+2.30i)T+(8.5−14.7i)T2 1 + (-3.99 + 2.30i)T + (8.5 - 14.7i)T^{2}
19 1+(2.52+0.675i)T+(16.4+9.5i)T2 1 + (2.52 + 0.675i)T + (16.4 + 9.5i)T^{2}
23 1+(−1.62+2.82i)T+(−11.5−19.9i)T2 1 + (-1.62 + 2.82i)T + (-11.5 - 19.9i)T^{2}
29 1+(−7.35−4.24i)T+(14.5+25.1i)T2 1 + (-7.35 - 4.24i)T + (14.5 + 25.1i)T^{2}
31 1+(2.29+2.29i)T+31iT2 1 + (2.29 + 2.29i)T + 31iT^{2}
37 1+(1.94+7.25i)T+(−32.0+18.5i)T2 1 + (1.94 + 7.25i)T + (-32.0 + 18.5i)T^{2}
41 1+(0.391+1.46i)T+(−35.5+20.5i)T2 1 + (0.391 + 1.46i)T + (-35.5 + 20.5i)T^{2}
43 1+(−9.48+5.47i)T+(21.5−37.2i)T2 1 + (-9.48 + 5.47i)T + (21.5 - 37.2i)T^{2}
47 1+(−3.31+3.31i)T−47iT2 1 + (-3.31 + 3.31i)T - 47iT^{2}
53 1−2.94iT−53T2 1 - 2.94iT - 53T^{2}
59 1+(0.115−0.432i)T+(−51.0−29.5i)T2 1 + (0.115 - 0.432i)T + (-51.0 - 29.5i)T^{2}
61 1+(9.70−5.60i)T+(30.5−52.8i)T2 1 + (9.70 - 5.60i)T + (30.5 - 52.8i)T^{2}
67 1+(2.48+9.27i)T+(−58.0+33.5i)T2 1 + (2.48 + 9.27i)T + (-58.0 + 33.5i)T^{2}
71 1+(−0.740+2.76i)T+(−61.4−35.5i)T2 1 + (-0.740 + 2.76i)T + (-61.4 - 35.5i)T^{2}
73 1+(−0.0928−0.0928i)T+73iT2 1 + (-0.0928 - 0.0928i)T + 73iT^{2}
79 1−9.86iT−79T2 1 - 9.86iT - 79T^{2}
83 1+(−7.31+7.31i)T−83iT2 1 + (-7.31 + 7.31i)T - 83iT^{2}
89 1+(−0.299+0.0801i)T+(77.0−44.5i)T2 1 + (-0.299 + 0.0801i)T + (77.0 - 44.5i)T^{2}
97 1+(−10.4−2.79i)T+(84.0+48.5i)T2 1 + (-10.4 - 2.79i)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

−11.04823725858822607369646022939, −10.41920653769642286780641991448, −9.211335582021300161942124886012, −8.694546916532034878357193294004, −7.51802109944824909244174218691, −6.20285775000030003852951985239, −5.15718228912658967021477227562, −4.21355349344332133793831872910, −3.52264078700980140143633112988, −0.75337283437286606855203255331, 1.34090913028294365264397895072, 3.20654746479519048605492654851, 4.18211051137595158519395161278, 6.11503426397286570506355882238, 6.52888165923534069801078321931, 7.36711846417605559124128374940, 8.248629509426752744620285507818, 9.507187117675753230510456349638, 10.67563056866435722994192540003, 11.49233747922562781589600924967

Graph of the ZZ-function along the critical line