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Properties

Label 2-416-32.21-c1-0-12
Degree 22
Conductor 416416
Sign 0.428−0.903i0.428 - 0.903i
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 + 0.152i)2-s + (−0.435 + 0.180i)3-s + (1.95 − 0.427i)4-s + (0.326 − 0.788i)5-s + (0.585 − 0.320i)6-s + (1.89 + 1.89i)7-s + (−2.68 + 0.898i)8-s + (−1.96 + 1.96i)9-s + (−0.339 + 1.15i)10-s + (1.20 + 0.500i)11-s + (−0.774 + 0.539i)12-s + (−0.382 − 0.923i)13-s + (−2.94 − 2.37i)14-s + 0.402i·15-s + (3.63 − 1.67i)16-s + 1.89i·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.107i)2-s + (−0.251 + 0.104i)3-s + (0.976 − 0.213i)4-s + (0.145 − 0.352i)5-s + (0.239 − 0.130i)6-s + (0.714 + 0.714i)7-s + (−0.948 + 0.317i)8-s + (−0.654 + 0.654i)9-s + (−0.107 + 0.366i)10-s + (0.364 + 0.150i)11-s + (−0.223 + 0.155i)12-s + (−0.106 − 0.256i)13-s + (−0.787 − 0.633i)14-s + 0.103i·15-s + (0.908 − 0.418i)16-s + 0.458i·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.428−0.903i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.428−0.903i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.428−0.903i0.428 - 0.903i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(53,⋅)\chi_{416} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.428−0.903i)(2,\ 416,\ (\ :1/2),\ 0.428 - 0.903i)

Particular Values

L(1)L(1) ≈\approx 0.699118+0.442272i0.699118 + 0.442272i
L(12)L(\frac12) ≈\approx 0.699118+0.442272i0.699118 + 0.442272i
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+(1.40−0.152i)T 1 + (1.40 - 0.152i)T
13 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
good3 1+(0.435−0.180i)T+(2.12−2.12i)T2 1 + (0.435 - 0.180i)T + (2.12 - 2.12i)T^{2}
5 1+(−0.326+0.788i)T+(−3.53−3.53i)T2 1 + (-0.326 + 0.788i)T + (-3.53 - 3.53i)T^{2}
7 1+(−1.89−1.89i)T+7iT2 1 + (-1.89 - 1.89i)T + 7iT^{2}
11 1+(−1.20−0.500i)T+(7.77+7.77i)T2 1 + (-1.20 - 0.500i)T + (7.77 + 7.77i)T^{2}
17 1−1.89iT−17T2 1 - 1.89iT - 17T^{2}
19 1+(−0.478−1.15i)T+(−13.4+13.4i)T2 1 + (-0.478 - 1.15i)T + (-13.4 + 13.4i)T^{2}
23 1+(0.331−0.331i)T−23iT2 1 + (0.331 - 0.331i)T - 23iT^{2}
29 1+(−0.842+0.349i)T+(20.5−20.5i)T2 1 + (-0.842 + 0.349i)T + (20.5 - 20.5i)T^{2}
31 1−6.24T+31T2 1 - 6.24T + 31T^{2}
37 1+(4.38−10.5i)T+(−26.1−26.1i)T2 1 + (4.38 - 10.5i)T + (-26.1 - 26.1i)T^{2}
41 1+(8.18−8.18i)T−41iT2 1 + (8.18 - 8.18i)T - 41iT^{2}
43 1+(−9.38−3.88i)T+(30.4+30.4i)T2 1 + (-9.38 - 3.88i)T + (30.4 + 30.4i)T^{2}
47 1−4.42iT−47T2 1 - 4.42iT - 47T^{2}
53 1+(−1.51−0.626i)T+(37.4+37.4i)T2 1 + (-1.51 - 0.626i)T + (37.4 + 37.4i)T^{2}
59 1+(−4.71+11.3i)T+(−41.7−41.7i)T2 1 + (-4.71 + 11.3i)T + (-41.7 - 41.7i)T^{2}
61 1+(−10.2+4.24i)T+(43.1−43.1i)T2 1 + (-10.2 + 4.24i)T + (43.1 - 43.1i)T^{2}
67 1+(0.969−0.401i)T+(47.3−47.3i)T2 1 + (0.969 - 0.401i)T + (47.3 - 47.3i)T^{2}
71 1+(−5.81−5.81i)T+71iT2 1 + (-5.81 - 5.81i)T + 71iT^{2}
73 1+(6.10−6.10i)T−73iT2 1 + (6.10 - 6.10i)T - 73iT^{2}
79 1+4.79iT−79T2 1 + 4.79iT - 79T^{2}
83 1+(4.15+10.0i)T+(−58.6+58.6i)T2 1 + (4.15 + 10.0i)T + (-58.6 + 58.6i)T^{2}
89 1+(2.78+2.78i)T+89iT2 1 + (2.78 + 2.78i)T + 89iT^{2}
97 1+15.0T+97T2 1 + 15.0T + 97T^{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

−11.38370985744105637745517607076, −10.38026376546088921599675710216, −9.544490982868037715850084926833, −8.358170603403198829930780284108, −8.213117755029727202933506287311, −6.75265325588550515244401823486, −5.71479160467744957991612749910, −4.87472708556401376884796608735, −2.85184991645459444407365153179, −1.52356058045961887022914078193, 0.814271679901212791286990174948, 2.47994573903582241138665026070, 3.87956793228244409625793902292, 5.51330234675057906639890013357, 6.68395375954733265753819140551, 7.27278892799193795072177567565, 8.472931262418226755235090412067, 9.135031652983562891653327181924, 10.28536411718911042932795765500, 10.90348539093745912603836948751

Graph of the ZZ-function along the critical line