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Properties

Label 2-416-416.99-c1-0-20
Degree 22
Conductor 416416
Sign 0.190−0.981i0.190 - 0.981i
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  + (−1.40 + 0.146i)2-s + (−0.514 + 1.24i)3-s + (1.95 − 0.411i)4-s + (−0.618 + 1.49i)5-s + (0.542 − 1.82i)6-s + 4.58·7-s + (−2.69 + 0.864i)8-s + (0.843 + 0.843i)9-s + (0.652 − 2.19i)10-s + (0.0872 + 0.0361i)11-s + (−0.496 + 2.64i)12-s + (0.783 − 3.51i)13-s + (−6.45 + 0.670i)14-s + (−1.53 − 1.53i)15-s + (3.66 − 1.60i)16-s + 2.55·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.103i)2-s + (−0.296 + 0.716i)3-s + (0.978 − 0.205i)4-s + (−0.276 + 0.668i)5-s + (0.221 − 0.743i)6-s + 1.73·7-s + (−0.952 + 0.305i)8-s + (0.281 + 0.281i)9-s + (0.206 − 0.693i)10-s + (0.0263 + 0.0108i)11-s + (−0.143 + 0.762i)12-s + (0.217 − 0.976i)13-s + (−1.72 + 0.179i)14-s + (−0.396 − 0.396i)15-s + (0.915 − 0.402i)16-s + 0.620·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.190−0.981i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.190−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.190−0.981i0.190 - 0.981i
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), 0.190−0.981i)(2,\ 416,\ (\ :1/2),\ 0.190 - 0.981i)

Particular Values

L(1)L(1) ≈\approx 0.749287+0.617793i0.749287 + 0.617793i
L(12)L(\frac12) ≈\approx 0.749287+0.617793i0.749287 + 0.617793i
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+(1.40−0.146i)T 1 + (1.40 - 0.146i)T
13 1+(−0.783+3.51i)T 1 + (-0.783 + 3.51i)T
good3 1+(0.514−1.24i)T+(−2.12−2.12i)T2 1 + (0.514 - 1.24i)T + (-2.12 - 2.12i)T^{2}
5 1+(0.618−1.49i)T+(−3.53−3.53i)T2 1 + (0.618 - 1.49i)T + (-3.53 - 3.53i)T^{2}
7 1−4.58T+7T2 1 - 4.58T + 7T^{2}
11 1+(−0.0872−0.0361i)T+(7.77+7.77i)T2 1 + (-0.0872 - 0.0361i)T + (7.77 + 7.77i)T^{2}
17 1−2.55T+17T2 1 - 2.55T + 17T^{2}
19 1+(0.456+1.10i)T+(−13.4+13.4i)T2 1 + (0.456 + 1.10i)T + (-13.4 + 13.4i)T^{2}
23 1+(−1.27−1.27i)T+23iT2 1 + (-1.27 - 1.27i)T + 23iT^{2}
29 1+(−0.966−0.400i)T+(20.5+20.5i)T2 1 + (-0.966 - 0.400i)T + (20.5 + 20.5i)T^{2}
31 1+(1.62−1.62i)T−31iT2 1 + (1.62 - 1.62i)T - 31iT^{2}
37 1+(−0.285−0.118i)T+(26.1+26.1i)T2 1 + (-0.285 - 0.118i)T + (26.1 + 26.1i)T^{2}
41 1−4.15iT−41T2 1 - 4.15iT - 41T^{2}
43 1+(8.33−3.45i)T+(30.4−30.4i)T2 1 + (8.33 - 3.45i)T + (30.4 - 30.4i)T^{2}
47 1+(6.32−6.32i)T−47iT2 1 + (6.32 - 6.32i)T - 47iT^{2}
53 1+(−3.50+1.45i)T+(37.4−37.4i)T2 1 + (-3.50 + 1.45i)T + (37.4 - 37.4i)T^{2}
59 1+(−5.30+12.8i)T+(−41.7−41.7i)T2 1 + (-5.30 + 12.8i)T + (-41.7 - 41.7i)T^{2}
61 1+(−7.13−2.95i)T+(43.1+43.1i)T2 1 + (-7.13 - 2.95i)T + (43.1 + 43.1i)T^{2}
67 1+(13.6−5.66i)T+(47.3−47.3i)T2 1 + (13.6 - 5.66i)T + (47.3 - 47.3i)T^{2}
71 1+0.284iT−71T2 1 + 0.284iT - 71T^{2}
73 1−1.57T+73T2 1 - 1.57T + 73T^{2}
79 1+3.48T+79T2 1 + 3.48T + 79T^{2}
83 1+(2.91+7.04i)T+(−58.6+58.6i)T2 1 + (2.91 + 7.04i)T + (-58.6 + 58.6i)T^{2}
89 1+14.6iT−89T2 1 + 14.6iT - 89T^{2}
97 1+(10.1+10.1i)T+97iT2 1 + (10.1 + 10.1i)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

−11.18179872767777914014034978087, −10.53357585549075223483653231109, −9.828850580537368417552386230697, −8.533095408710668098754533173597, −7.85427488044653708172902484389, −7.09030107794876489760638598041, −5.62450371911466080374409584828, −4.76450543754327888174678980288, −3.17702757026756942737211038519, −1.54338735709664151633885829392, 1.06542663758188521434020761208, 1.95791999781993410803455387247, 4.07134693487118298311564053588, 5.34320062218775638240695823749, 6.63663376773082590756317024190, 7.48935520477045794126026952953, 8.307584868350665478992367105087, 8.918221507796646419976981168430, 10.12682953622671667115268570225, 11.14913354408919950124928334847

Graph of the ZZ-function along the critical line