Properties

Label 2-416-416.99-c1-0-8
Degree 22
Conductor 416416
Sign 0.9840.177i0.984 - 0.177i
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  + (−0.717 − 1.21i)2-s + (0.172 − 0.415i)3-s + (−0.969 + 1.74i)4-s + (−0.252 + 0.610i)5-s + (−0.629 + 0.0885i)6-s − 3.40·7-s + (2.82 − 0.0751i)8-s + (1.97 + 1.97i)9-s + (0.925 − 0.130i)10-s + (−1.70 − 0.704i)11-s + (0.560 + 0.703i)12-s + (3.28 + 1.47i)13-s + (2.44 + 4.14i)14-s + (0.210 + 0.210i)15-s + (−2.12 − 3.39i)16-s + 2.30·17-s + ⋯
L(s)  = 1  + (−0.507 − 0.861i)2-s + (0.0993 − 0.239i)3-s + (−0.484 + 0.874i)4-s + (−0.113 + 0.272i)5-s + (−0.257 + 0.0361i)6-s − 1.28·7-s + (0.999 − 0.0265i)8-s + (0.659 + 0.659i)9-s + (0.292 − 0.0411i)10-s + (−0.512 − 0.212i)11-s + (0.161 + 0.203i)12-s + (0.912 + 0.410i)13-s + (0.652 + 1.10i)14-s + (0.0542 + 0.0542i)15-s + (−0.530 − 0.847i)16-s + 0.559·17-s + ⋯

Functional equation

Λ(s)=(416s/2ΓC(s)L(s)=((0.9840.177i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2ΓC(s+1/2)L(s)=((0.9840.177i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25132^{5} \cdot 13
Sign: 0.9840.177i0.984 - 0.177i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(99,)\chi_{416} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.9840.177i)(2,\ 416,\ (\ :1/2),\ 0.984 - 0.177i)

Particular Values

L(1)L(1) \approx 0.877264+0.0785839i0.877264 + 0.0785839i
L(12)L(\frac12) \approx 0.877264+0.0785839i0.877264 + 0.0785839i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.717+1.21i)T 1 + (0.717 + 1.21i)T
13 1+(3.281.47i)T 1 + (-3.28 - 1.47i)T
good3 1+(0.172+0.415i)T+(2.122.12i)T2 1 + (-0.172 + 0.415i)T + (-2.12 - 2.12i)T^{2}
5 1+(0.2520.610i)T+(3.533.53i)T2 1 + (0.252 - 0.610i)T + (-3.53 - 3.53i)T^{2}
7 1+3.40T+7T2 1 + 3.40T + 7T^{2}
11 1+(1.70+0.704i)T+(7.77+7.77i)T2 1 + (1.70 + 0.704i)T + (7.77 + 7.77i)T^{2}
17 12.30T+17T2 1 - 2.30T + 17T^{2}
19 1+(2.927.05i)T+(13.4+13.4i)T2 1 + (-2.92 - 7.05i)T + (-13.4 + 13.4i)T^{2}
23 1+(1.961.96i)T+23iT2 1 + (-1.96 - 1.96i)T + 23iT^{2}
29 1+(0.8200.339i)T+(20.5+20.5i)T2 1 + (-0.820 - 0.339i)T + (20.5 + 20.5i)T^{2}
31 1+(1.781.78i)T31iT2 1 + (1.78 - 1.78i)T - 31iT^{2}
37 1+(6.76+2.80i)T+(26.1+26.1i)T2 1 + (6.76 + 2.80i)T + (26.1 + 26.1i)T^{2}
41 15.14iT41T2 1 - 5.14iT - 41T^{2}
43 1+(0.493+0.204i)T+(30.430.4i)T2 1 + (-0.493 + 0.204i)T + (30.4 - 30.4i)T^{2}
47 1+(1.59+1.59i)T47iT2 1 + (-1.59 + 1.59i)T - 47iT^{2}
53 1+(5.87+2.43i)T+(37.437.4i)T2 1 + (-5.87 + 2.43i)T + (37.4 - 37.4i)T^{2}
59 1+(2.54+6.15i)T+(41.741.7i)T2 1 + (-2.54 + 6.15i)T + (-41.7 - 41.7i)T^{2}
61 1+(5.74+2.37i)T+(43.1+43.1i)T2 1 + (5.74 + 2.37i)T + (43.1 + 43.1i)T^{2}
67 1+(2.691.11i)T+(47.347.3i)T2 1 + (2.69 - 1.11i)T + (47.3 - 47.3i)T^{2}
71 17.20iT71T2 1 - 7.20iT - 71T^{2}
73 16.22T+73T2 1 - 6.22T + 73T^{2}
79 1+16.4T+79T2 1 + 16.4T + 79T^{2}
83 1+(0.3840.929i)T+(58.6+58.6i)T2 1 + (-0.384 - 0.929i)T + (-58.6 + 58.6i)T^{2}
89 19.50iT89T2 1 - 9.50iT - 89T^{2}
97 1+(8.56+8.56i)T+97iT2 1 + (8.56 + 8.56i)T + 97iT^{2}
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   L(s)=p j=12(1αj,pps)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

−11.06169802841131362449239928102, −10.29169314916517795929867509054, −9.673991436024970400872308925928, −8.614418185004196079466429694811, −7.65644860739699845483178208165, −6.82469844329005342880997745907, −5.42637643006964263883042536029, −3.80713995409934219012046779173, −3.06842454212734969433620097236, −1.48570281467204913465669196117, 0.74365433081774179191023450742, 3.15146565932193268372512634530, 4.46697846270765032087950026493, 5.62681198963195930626749384267, 6.66393708526679060106879105453, 7.30149667870479653845565627372, 8.626567421799399314674416570286, 9.235355794820571090033356016303, 10.07387129274528481152397085944, 10.75509591270612247399470493642

Graph of the ZZ-function along the critical line