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Properties

Label 2-630-5.4-c1-0-3
Degree 22
Conductor 630630
Sign 0.994+0.100i0.994 + 0.100i
Analytic cond. 5.030575.03057
Root an. cond. 2.242892.24289
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  − i·2-s − 4-s + (−2.22 − 0.224i)5-s + i·7-s + i·8-s + (−0.224 + 2.22i)10-s + 4.89·11-s + 4.44i·13-s + 14-s + 16-s + 2i·17-s − 1.55·19-s + (2.22 + 0.224i)20-s − 4.89i·22-s − 2.89i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.994 − 0.100i)5-s + 0.377i·7-s + 0.353i·8-s + (−0.0710 + 0.703i)10-s + 1.47·11-s + 1.23i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s − 0.355·19-s + (0.497 + 0.0502i)20-s − 1.04i·22-s − 0.604i·23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((0.994+0.100i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+1/2) L(s)=((0.994+0.100i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.994+0.100i0.994 + 0.100i
Analytic conductor: 5.030575.03057
Root analytic conductor: 2.242892.24289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ630(379,⋅)\chi_{630} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :1/2), 0.994+0.100i)(2,\ 630,\ (\ :1/2),\ 0.994 + 0.100i)

Particular Values

L(1)L(1) ≈\approx 1.20180−0.0605493i1.20180 - 0.0605493i
L(12)L(\frac12) ≈\approx 1.20180−0.0605493i1.20180 - 0.0605493i
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+iT 1 + iT
3 1 1
5 1+(2.22+0.224i)T 1 + (2.22 + 0.224i)T
7 1−iT 1 - iT
good11 1−4.89T+11T2 1 - 4.89T + 11T^{2}
13 1−4.44iT−13T2 1 - 4.44iT - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+1.55T+19T2 1 + 1.55T + 19T^{2}
23 1+2.89iT−23T2 1 + 2.89iT - 23T^{2}
29 1−6.89T+29T2 1 - 6.89T + 29T^{2}
31 1−8.89T+31T2 1 - 8.89T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−1.10T+41T2 1 - 1.10T + 41T^{2}
43 1+0.898iT−43T2 1 + 0.898iT - 43T^{2}
47 1−8.89iT−47T2 1 - 8.89iT - 47T^{2}
53 1−10.8iT−53T2 1 - 10.8iT - 53T^{2}
59 1+1.55T+59T2 1 + 1.55T + 59T^{2}
61 1−3.55T+61T2 1 - 3.55T + 61T^{2}
67 1−8iT−67T2 1 - 8iT - 67T^{2}
71 1−1.10T+71T2 1 - 1.10T + 71T^{2}
73 1−2.89iT−73T2 1 - 2.89iT - 73T^{2}
79 1+6.89T+79T2 1 + 6.89T + 79T^{2}
83 1−2.44iT−83T2 1 - 2.44iT - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+15.7iT−97T2 1 + 15.7iT - 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

−10.77978238130661553106619261712, −9.688977376503746855911810381446, −8.830133923242571812937321381777, −8.305228659001904633933983976304, −6.96789027740596121622536605099, −6.17813385027328162864953002477, −4.47739934615752819725415685874, −4.13835302570792377367435308883, −2.78941120750453085865894528983, −1.26269572151845981901787625124, 0.822428553646264225555685138781, 3.17361220245073802465020795395, 4.11105245719994294565629021426, 5.04091120199844640166316057104, 6.37534852595493371583771746251, 7.00375233718295929533320522989, 8.008271270560157847940488043327, 8.557246996831482733398238683821, 9.679831163742516411236608743851, 10.52249599462199763648347994259

Graph of the ZZ-function along the critical line