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Properties

Label 2-416-416.77-c1-0-28
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(77,⋅)\chi_{416} (77, \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

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

Graph of the ZZ-function along the critical line