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Properties

Label 2-702-13.3-c1-0-12
Degree 22
Conductor 702702
Sign 0.964−0.265i0.964 - 0.265i
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 2.30·5-s + (1.65 − 2.86i)7-s − 0.999·8-s + (1.15 + 1.99i)10-s + (−0.348 − 0.603i)11-s + (1.80 − 3.12i)13-s + 3.30·14-s + (−0.5 − 0.866i)16-s + (1.95 − 3.38i)17-s + (−0.197 + 0.341i)19-s + (−1.15 + 1.99i)20-s + (0.348 − 0.603i)22-s + (0.802 + 1.39i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.02·5-s + (0.624 − 1.08i)7-s − 0.353·8-s + (0.364 + 0.630i)10-s + (−0.105 − 0.182i)11-s + (0.499 − 0.866i)13-s + 0.882·14-s + (−0.125 − 0.216i)16-s + (0.473 − 0.820i)17-s + (−0.0452 + 0.0783i)19-s + (−0.257 + 0.445i)20-s + (0.0743 − 0.128i)22-s + (0.167 + 0.289i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((0.964−0.265i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((0.964−0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 0.964−0.265i0.964 - 0.265i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(55,⋅)\chi_{702} (55, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), 0.964−0.265i)(2,\ 702,\ (\ :1/2),\ 0.964 - 0.265i)

Particular Values

L(1)L(1) ≈\approx 2.20937+0.298078i2.20937 + 0.298078i
L(12)L(\frac12) ≈\approx 2.20937+0.298078i2.20937 + 0.298078i
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1 1
13 1+(−1.80+3.12i)T 1 + (-1.80 + 3.12i)T
good5 1−2.30T+5T2 1 - 2.30T + 5T^{2}
7 1+(−1.65+2.86i)T+(−3.5−6.06i)T2 1 + (-1.65 + 2.86i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.348+0.603i)T+(−5.5+9.52i)T2 1 + (0.348 + 0.603i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.95+3.38i)T+(−8.5−14.7i)T2 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.197−0.341i)T+(−9.5−16.4i)T2 1 + (0.197 - 0.341i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.802−1.39i)T+(−11.5+19.9i)T2 1 + (-0.802 - 1.39i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.80−6.58i)T+(−14.5+25.1i)T2 1 + (-3.80 - 6.58i)T + (-14.5 + 25.1i)T^{2}
31 1+7.21T+31T2 1 + 7.21T + 31T^{2}
37 1+(3.65+6.32i)T+(−18.5+32.0i)T2 1 + (3.65 + 6.32i)T + (-18.5 + 32.0i)T^{2}
41 1+(−5.40−9.36i)T+(−20.5+35.5i)T2 1 + (-5.40 - 9.36i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.302−0.524i)T+(−21.5−37.2i)T2 1 + (0.302 - 0.524i)T + (-21.5 - 37.2i)T^{2}
47 1−4.60T+47T2 1 - 4.60T + 47T^{2}
53 1−3T+53T2 1 - 3T + 53T^{2}
59 1+(7.15−12.3i)T+(−29.5−51.0i)T2 1 + (7.15 - 12.3i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.84−4.93i)T+(−30.5−52.8i)T2 1 + (2.84 - 4.93i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.15−5.45i)T+(−33.5+58.0i)T2 1 + (-3.15 - 5.45i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.34−5.79i)T+(−35.5−61.4i)T2 1 + (3.34 - 5.79i)T + (-35.5 - 61.4i)T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1+1.90T+79T2 1 + 1.90T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(8.40+14.5i)T+(−44.5+77.0i)T2 1 + (8.40 + 14.5i)T + (-44.5 + 77.0i)T^{2}
97 1+(4.90−8.50i)T+(−48.5−84.0i)T2 1 + (4.90 - 8.50i)T + (-48.5 - 84.0i)T^{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

−10.51704440999512226450231814508, −9.598248352773422311182738289456, −8.674739355117514559701129834657, −7.65474017058673662464117824566, −7.04978983255978034289662667015, −5.83683665687215791952549977896, −5.27786471653958429536602100601, −4.13405019361891599841001095803, −2.94295667457664057063841697905, −1.24149652574381936140579056937, 1.69428987950209548240702949921, 2.36715786900079416652273695234, 3.83411503509831759783413542754, 5.02963971157197880670249197722, 5.78118850307618146031729945867, 6.53038239759625173192531350412, 8.036264461569206541386794302055, 8.942627431177712714831083123820, 9.552934366082786627613081045250, 10.49580853129725126603180970912

Graph of the ZZ-function along the critical line