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Properties

Label 2-416-416.395-c1-0-41
Degree 22
Conductor 416416
Sign −0.951+0.307i-0.951 + 0.307i
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  + (−1.41 + 0.0124i)2-s + (0.185 + 0.447i)3-s + (1.99 − 0.0352i)4-s + (−0.172 − 0.416i)5-s + (−0.267 − 0.630i)6-s − 2.63·7-s + (−2.82 + 0.0748i)8-s + (1.95 − 1.95i)9-s + (0.249 + 0.586i)10-s + (−3.87 + 1.60i)11-s + (0.386 + 0.888i)12-s + (−3.51 − 0.795i)13-s + (3.72 − 0.0328i)14-s + (0.154 − 0.154i)15-s + (3.99 − 0.141i)16-s − 3.57·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00882i)2-s + (0.107 + 0.258i)3-s + (0.999 − 0.0176i)4-s + (−0.0771 − 0.186i)5-s + (−0.109 − 0.257i)6-s − 0.995·7-s + (−0.999 + 0.0264i)8-s + (0.651 − 0.651i)9-s + (0.0787 + 0.185i)10-s + (−1.16 + 0.484i)11-s + (0.111 + 0.256i)12-s + (−0.975 − 0.220i)13-s + (0.995 − 0.00878i)14-s + (0.0398 − 0.0398i)15-s + (0.999 − 0.0352i)16-s − 0.867·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.951+0.307i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.951+0.307i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.951+0.307i-0.951 + 0.307i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(395,⋅)\chi_{416} (395, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.951+0.307i)(2,\ 416,\ (\ :1/2),\ -0.951 + 0.307i)

Particular Values

L(1)L(1) ≈\approx 0.0145952−0.0927568i0.0145952 - 0.0927568i
L(12)L(\frac12) ≈\approx 0.0145952−0.0927568i0.0145952 - 0.0927568i
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+(1.41−0.0124i)T 1 + (1.41 - 0.0124i)T
13 1+(3.51+0.795i)T 1 + (3.51 + 0.795i)T
good3 1+(−0.185−0.447i)T+(−2.12+2.12i)T2 1 + (-0.185 - 0.447i)T + (-2.12 + 2.12i)T^{2}
5 1+(0.172+0.416i)T+(−3.53+3.53i)T2 1 + (0.172 + 0.416i)T + (-3.53 + 3.53i)T^{2}
7 1+2.63T+7T2 1 + 2.63T + 7T^{2}
11 1+(3.87−1.60i)T+(7.77−7.77i)T2 1 + (3.87 - 1.60i)T + (7.77 - 7.77i)T^{2}
17 1+3.57T+17T2 1 + 3.57T + 17T^{2}
19 1+(0.249−0.603i)T+(−13.4−13.4i)T2 1 + (0.249 - 0.603i)T + (-13.4 - 13.4i)T^{2}
23 1+(2.69−2.69i)T−23iT2 1 + (2.69 - 2.69i)T - 23iT^{2}
29 1+(1.95−0.808i)T+(20.5−20.5i)T2 1 + (1.95 - 0.808i)T + (20.5 - 20.5i)T^{2}
31 1+(3.43+3.43i)T+31iT2 1 + (3.43 + 3.43i)T + 31iT^{2}
37 1+(2.19−0.907i)T+(26.1−26.1i)T2 1 + (2.19 - 0.907i)T + (26.1 - 26.1i)T^{2}
41 1−0.550iT−41T2 1 - 0.550iT - 41T^{2}
43 1+(4.93+2.04i)T+(30.4+30.4i)T2 1 + (4.93 + 2.04i)T + (30.4 + 30.4i)T^{2}
47 1+(2.63+2.63i)T+47iT2 1 + (2.63 + 2.63i)T + 47iT^{2}
53 1+(11.0+4.58i)T+(37.4+37.4i)T2 1 + (11.0 + 4.58i)T + (37.4 + 37.4i)T^{2}
59 1+(0.722+1.74i)T+(−41.7+41.7i)T2 1 + (0.722 + 1.74i)T + (-41.7 + 41.7i)T^{2}
61 1+(−2.34+0.973i)T+(43.1−43.1i)T2 1 + (-2.34 + 0.973i)T + (43.1 - 43.1i)T^{2}
67 1+(−10.2−4.23i)T+(47.3+47.3i)T2 1 + (-10.2 - 4.23i)T + (47.3 + 47.3i)T^{2}
71 1−0.852iT−71T2 1 - 0.852iT - 71T^{2}
73 1−6.22T+73T2 1 - 6.22T + 73T^{2}
79 1+1.67T+79T2 1 + 1.67T + 79T^{2}
83 1+(3.20−7.73i)T+(−58.6−58.6i)T2 1 + (3.20 - 7.73i)T + (-58.6 - 58.6i)T^{2}
89 1−3.24iT−89T2 1 - 3.24iT - 89T^{2}
97 1+(−13.6+13.6i)T−97iT2 1 + (-13.6 + 13.6i)T - 97iT^{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

−10.37378084258214575161844734619, −9.879504411165509557723557535463, −9.218664100042432537500691007969, −8.096685403730316685307567256924, −7.16689124799014682962208946087, −6.39876696096767270380128798445, −5.01814173886015846097775511263, −3.48714348112842098169378105048, −2.23588601723039665499962157368, −0.07380022883093821362308396263, 2.12015535887364668814730390502, 3.15222514264820210374902602688, 4.97078178421053723326455733696, 6.35207339830799185622300587889, 7.15661789449653373788506057003, 7.893695008413628642795489632245, 8.930631109901544450764773888446, 9.868196627446215161660033462273, 10.53453925001558907679997214274, 11.29834095592125375872089879701

Graph of the ZZ-function along the critical line