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Properties

Label 2-490-7.2-c3-0-6
Degree 22
Conductor 490490
Sign −0.386−0.922i-0.386 - 0.922i
Analytic cond. 28.910928.9109
Root an. cond. 5.376885.37688
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (−2.5 + 4.33i)5-s + 6·6-s − 7.99·8-s + (9 − 15.5i)9-s + (5 + 8.66i)10-s + (8.5 + 14.7i)11-s + (6.00 − 10.3i)12-s − 81·13-s − 15.0·15-s + (−8 + 13.8i)16-s + (45.5 + 78.8i)17-s + (−18 − 31.1i)18-s + (−51 + 88.3i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s − 0.353·8-s + (0.333 − 0.577i)9-s + (0.158 + 0.273i)10-s + (0.232 + 0.403i)11-s + (0.144 − 0.249i)12-s − 1.72·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.649 + 1.12i)17-s + (−0.235 − 0.408i)18-s + (−0.615 + 1.06i)19-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=((−0.386−0.922i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+3/2) L(s)=((−0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 490490    =    2⋅5⋅722 \cdot 5 \cdot 7^{2}
Sign: −0.386−0.922i-0.386 - 0.922i
Analytic conductor: 28.910928.9109
Root analytic conductor: 5.376885.37688
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ490(471,⋅)\chi_{490} (471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 490, ( :3/2), −0.386−0.922i)(2,\ 490,\ (\ :3/2),\ -0.386 - 0.922i)

Particular Values

L(2)L(2) ≈\approx 1.0028806581.002880658
L(12)L(\frac12) ≈\approx 1.0028806581.002880658
L(52)L(\frac{5}{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+1.73i)T 1 + (-1 + 1.73i)T
5 1+(2.5−4.33i)T 1 + (2.5 - 4.33i)T
7 1 1
good3 1+(−1.5−2.59i)T+(−13.5+23.3i)T2 1 + (-1.5 - 2.59i)T + (-13.5 + 23.3i)T^{2}
11 1+(−8.5−14.7i)T+(−665.5+1.15e3i)T2 1 + (-8.5 - 14.7i)T + (-665.5 + 1.15e3i)T^{2}
13 1+81T+2.19e3T2 1 + 81T + 2.19e3T^{2}
17 1+(−45.5−78.8i)T+(−2.45e3+4.25e3i)T2 1 + (-45.5 - 78.8i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(51−88.3i)T+(−3.42e3−5.94e3i)T2 1 + (51 - 88.3i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−45+77.9i)T+(−6.08e3−1.05e4i)T2 1 + (-45 + 77.9i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+129T+2.43e4T2 1 + 129T + 2.43e4T^{2}
31 1+(58+100.i)T+(−1.48e4+2.57e4i)T2 1 + (58 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(157−271.i)T+(−2.53e4−4.38e4i)T2 1 + (157 - 271. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+124T+6.89e4T2 1 + 124T + 6.89e4T^{2}
43 1+434T+7.95e4T2 1 + 434T + 7.95e4T^{2}
47 1+(248.5−430.i)T+(−5.19e4−8.99e4i)T2 1 + (248.5 - 430. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−292−505.i)T+(−7.44e4+1.28e5i)T2 1 + (-292 - 505. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−166−287.i)T+(−1.02e5+1.77e5i)T2 1 + (-166 - 287. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(110−190.i)T+(−1.13e5−1.96e5i)T2 1 + (110 - 190. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(192+332.i)T+(−1.50e5+2.60e5i)T2 1 + (192 + 332. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+664T+3.57e5T2 1 + 664T + 3.57e5T^{2}
73 1+(115+199.i)T+(−1.94e5+3.36e5i)T2 1 + (115 + 199. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(180.5−312.i)T+(−2.46e5−4.26e5i)T2 1 + (180.5 - 312. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−1.17e3T+5.71e5T2 1 - 1.17e3T + 5.71e5T^{2}
89 1+(20−34.6i)T+(−3.52e5−6.10e5i)T2 1 + (20 - 34.6i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+175T+9.12e5T2 1 + 175T + 9.12e5T^{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

−10.57008077507323082531613537551, −10.09763049182849821061936742125, −9.379049872101267325123945917934, −8.236397924584690690118901958643, −7.14452600246217062658953551468, −6.10780251559037002648656274849, −4.79712136615398120286490716957, −3.96462540800582875329562099558, −3.00826073633262746989049859498, −1.67905153712234932726601862466, 0.25284674047600444708128237707, 2.07461690951247887983695099257, 3.35778008061874873853911664770, 4.85088696188358942287210006021, 5.28383253281939190261231757165, 7.02088306134278847687574229716, 7.23690921335504237567878646511, 8.292144243457909312595558124102, 9.176855228990429572957950978137, 10.10655963713706728285976052855

Graph of the ZZ-function along the critical line