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Properties

Label 2-416-416.389-c1-0-19
Degree 22
Conductor 416416
Sign 0.982−0.186i0.982 - 0.186i
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  + (−0.988 − 1.01i)2-s + (0.708 + 1.71i)3-s + (−0.0474 + 1.99i)4-s + (0.188 − 0.454i)5-s + (1.03 − 2.40i)6-s + (0.461 − 0.461i)7-s + (2.06 − 1.92i)8-s + (−0.301 + 0.301i)9-s + (−0.645 + 0.258i)10-s + (1.38 + 0.572i)11-s + (−3.45 + 1.33i)12-s + (2.41 − 2.67i)13-s + (−0.923 − 0.0109i)14-s + 0.910·15-s + (−3.99 − 0.189i)16-s + 1.70i·17-s + ⋯
L(s)  = 1  + (−0.698 − 0.715i)2-s + (0.408 + 0.987i)3-s + (−0.0237 + 0.999i)4-s + (0.0842 − 0.203i)5-s + (0.420 − 0.982i)6-s + (0.174 − 0.174i)7-s + (0.731 − 0.681i)8-s + (−0.100 + 0.100i)9-s + (−0.204 + 0.0817i)10-s + (0.416 + 0.172i)11-s + (−0.996 + 0.385i)12-s + (0.670 − 0.741i)13-s + (−0.246 − 0.00292i)14-s + 0.235·15-s + (−0.998 − 0.0474i)16-s + 0.413i·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.982−0.186i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.982−0.186i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.982−0.186i0.982 - 0.186i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(389,⋅)\chi_{416} (389, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.982−0.186i)(2,\ 416,\ (\ :1/2),\ 0.982 - 0.186i)

Particular Values

L(1)L(1) ≈\approx 1.22214+0.114886i1.22214 + 0.114886i
L(12)L(\frac12) ≈\approx 1.22214+0.114886i1.22214 + 0.114886i
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+(0.988+1.01i)T 1 + (0.988 + 1.01i)T
13 1+(−2.41+2.67i)T 1 + (-2.41 + 2.67i)T
good3 1+(−0.708−1.71i)T+(−2.12+2.12i)T2 1 + (-0.708 - 1.71i)T + (-2.12 + 2.12i)T^{2}
5 1+(−0.188+0.454i)T+(−3.53−3.53i)T2 1 + (-0.188 + 0.454i)T + (-3.53 - 3.53i)T^{2}
7 1+(−0.461+0.461i)T−7iT2 1 + (-0.461 + 0.461i)T - 7iT^{2}
11 1+(−1.38−0.572i)T+(7.77+7.77i)T2 1 + (-1.38 - 0.572i)T + (7.77 + 7.77i)T^{2}
17 1−1.70iT−17T2 1 - 1.70iT - 17T^{2}
19 1+(−1.71−4.14i)T+(−13.4+13.4i)T2 1 + (-1.71 - 4.14i)T + (-13.4 + 13.4i)T^{2}
23 1+(0.296−0.296i)T−23iT2 1 + (0.296 - 0.296i)T - 23iT^{2}
29 1+(0.673+1.62i)T+(−20.5+20.5i)T2 1 + (0.673 + 1.62i)T + (-20.5 + 20.5i)T^{2}
31 1−5.90iT−31T2 1 - 5.90iT - 31T^{2}
37 1+(1.12−2.70i)T+(−26.1−26.1i)T2 1 + (1.12 - 2.70i)T + (-26.1 - 26.1i)T^{2}
41 1+(6.52+6.52i)T+41iT2 1 + (6.52 + 6.52i)T + 41iT^{2}
43 1+(0.359−0.867i)T+(−30.4−30.4i)T2 1 + (0.359 - 0.867i)T + (-30.4 - 30.4i)T^{2}
47 1−2.98T+47T2 1 - 2.98T + 47T^{2}
53 1+(−4.36+10.5i)T+(−37.4−37.4i)T2 1 + (-4.36 + 10.5i)T + (-37.4 - 37.4i)T^{2}
59 1+(−3.83+9.25i)T+(−41.7−41.7i)T2 1 + (-3.83 + 9.25i)T + (-41.7 - 41.7i)T^{2}
61 1+(3.18+7.69i)T+(−43.1+43.1i)T2 1 + (3.18 + 7.69i)T + (-43.1 + 43.1i)T^{2}
67 1+(3.91−1.61i)T+(47.3−47.3i)T2 1 + (3.91 - 1.61i)T + (47.3 - 47.3i)T^{2}
71 1+(−3.34+3.34i)T−71iT2 1 + (-3.34 + 3.34i)T - 71iT^{2}
73 1+(7.51+7.51i)T+73iT2 1 + (7.51 + 7.51i)T + 73iT^{2}
79 1+2.03iT−79T2 1 + 2.03iT - 79T^{2}
83 1+(−0.942−2.27i)T+(−58.6+58.6i)T2 1 + (-0.942 - 2.27i)T + (-58.6 + 58.6i)T^{2}
89 1+(8.61−8.61i)T−89iT2 1 + (8.61 - 8.61i)T - 89iT^{2}
97 1−12.2iT−97T2 1 - 12.2iT - 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

−10.86186295799140824890276793147, −10.31564953504057592368739429077, −9.521851260919626720520068105664, −8.735241869702819604437237956362, −7.993387877146409583040427118531, −6.74534173821553678015793080185, −5.14902960430581636287445597482, −3.91232153343888182658347863929, −3.25760406126560837113114167055, −1.44188101091527142379886141606, 1.21323183950682249797627040483, 2.50758024008050730119770070165, 4.49506712798101561178187196677, 5.85355052047600230891143403635, 6.82958385175513779912473916956, 7.33459095602722421818548040953, 8.470212735133976992246101536012, 8.987658755549027064281887443683, 10.09929704853876231230530183202, 11.15127337385058209273504429190

Graph of the ZZ-function along the critical line