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Properties

Label 2-464-116.99-c1-0-6
Degree 22
Conductor 464464
Sign 0.672−0.739i0.672 - 0.739i
Analytic cond. 3.705053.70505
Root an. cond. 1.924851.92485
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  + (1.30 − 1.30i)3-s + 3.62i·5-s + 1.31i·7-s − 0.406i·9-s + (−0.158 + 0.158i)11-s + 4.10i·13-s + (4.73 + 4.73i)15-s + (−1.97 − 1.97i)17-s + (0.759 − 0.759i)19-s + (1.72 + 1.72i)21-s + 3.15i·23-s − 8.13·25-s + (3.38 + 3.38i)27-s + (−0.639 − 5.34i)29-s + (6.58 − 6.58i)31-s + ⋯
L(s)  = 1  + (0.753 − 0.753i)3-s + 1.62i·5-s + 0.498i·7-s − 0.135i·9-s + (−0.0478 + 0.0478i)11-s + 1.13i·13-s + (1.22 + 1.22i)15-s + (−0.479 − 0.479i)17-s + (0.174 − 0.174i)19-s + (0.375 + 0.375i)21-s + 0.658i·23-s − 1.62·25-s + (0.651 + 0.651i)27-s + (−0.118 − 0.992i)29-s + (1.18 − 1.18i)31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((0.672−0.739i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1/2) L(s)=((0.672−0.739i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 0.672−0.739i0.672 - 0.739i
Analytic conductor: 3.705053.70505
Root analytic conductor: 1.924851.92485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ464(447,⋅)\chi_{464} (447, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1/2), 0.672−0.739i)(2,\ 464,\ (\ :1/2),\ 0.672 - 0.739i)

Particular Values

L(1)L(1) ≈\approx 1.57195+0.695221i1.57195 + 0.695221i
L(12)L(\frac12) ≈\approx 1.57195+0.695221i1.57195 + 0.695221i
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
29 1+(0.639+5.34i)T 1 + (0.639 + 5.34i)T
good3 1+(−1.30+1.30i)T−3iT2 1 + (-1.30 + 1.30i)T - 3iT^{2}
5 1−3.62iT−5T2 1 - 3.62iT - 5T^{2}
7 1−1.31iT−7T2 1 - 1.31iT - 7T^{2}
11 1+(0.158−0.158i)T−11iT2 1 + (0.158 - 0.158i)T - 11iT^{2}
13 1−4.10iT−13T2 1 - 4.10iT - 13T^{2}
17 1+(1.97+1.97i)T+17iT2 1 + (1.97 + 1.97i)T + 17iT^{2}
19 1+(−0.759+0.759i)T−19iT2 1 + (-0.759 + 0.759i)T - 19iT^{2}
23 1−3.15iT−23T2 1 - 3.15iT - 23T^{2}
31 1+(−6.58+6.58i)T−31iT2 1 + (-6.58 + 6.58i)T - 31iT^{2}
37 1+(−0.338+0.338i)T−37iT2 1 + (-0.338 + 0.338i)T - 37iT^{2}
41 1+(−0.459+0.459i)T−41iT2 1 + (-0.459 + 0.459i)T - 41iT^{2}
43 1+(−7.46+7.46i)T−43iT2 1 + (-7.46 + 7.46i)T - 43iT^{2}
47 1+(4.11+4.11i)T+47iT2 1 + (4.11 + 4.11i)T + 47iT^{2}
53 1−7.59T+53T2 1 - 7.59T + 53T^{2}
59 1+10.0iT−59T2 1 + 10.0iT - 59T^{2}
61 1+(−3.51−3.51i)T+61iT2 1 + (-3.51 - 3.51i)T + 61iT^{2}
67 1−2.87T+67T2 1 - 2.87T + 67T^{2}
71 1+14.3T+71T2 1 + 14.3T + 71T^{2}
73 1+(8.40−8.40i)T−73iT2 1 + (8.40 - 8.40i)T - 73iT^{2}
79 1+(8.03−8.03i)T−79iT2 1 + (8.03 - 8.03i)T - 79iT^{2}
83 1−15.0iT−83T2 1 - 15.0iT - 83T^{2}
89 1+(7.68+7.68i)T+89iT2 1 + (7.68 + 7.68i)T + 89iT^{2}
97 1+(−3.15+3.15i)T−97iT2 1 + (-3.15 + 3.15i)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

−11.36770113396214843013146996959, −10.23878328291670734549092328227, −9.339869889250908775918466375284, −8.328985661609584039141049891103, −7.30390404255031212831566829916, −6.86466373433249160767371552828, −5.79023292749647554447685493735, −4.09840716860821675139893128836, −2.73806783289357424014306938547, −2.15843232149626265135839141059, 1.05365900026746372178944733465, 3.00808785884679229381246769654, 4.20702246067143134687363386681, 4.87182821043673754257108892945, 6.05213061868118060292366966498, 7.58878730106798418212474650062, 8.610123205406055368307273129566, 8.862435256468515798365169927673, 9.994826664867995722067383689536, 10.58811376381884648640130280528

Graph of the ZZ-function along the critical line