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Properties

Label 2-15e3-3.2-c0-0-0
Degree 22
Conductor 33753375
Sign 11
Analytic cond. 1.684341.68434
Root an. cond. 1.297821.29782
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82i·2-s − 2.33·4-s + 2.44i·8-s + 2.12·16-s + 1.95i·17-s − 1.82·19-s + 1.33i·23-s − 1.95·31-s − 1.44i·32-s + 3.57·34-s + 3.33i·38-s + 2.44·46-s + 1.61i·47-s − 49-s − 0.209i·53-s + ⋯
L(s)  = 1  − 1.82i·2-s − 2.33·4-s + 2.44i·8-s + 2.12·16-s + 1.95i·17-s − 1.82·19-s + 1.33i·23-s − 1.95·31-s − 1.44i·32-s + 3.57·34-s + 3.33i·38-s + 2.44·46-s + 1.61i·47-s − 49-s − 0.209i·53-s + ⋯

Functional equation

Λ(s)=(3375s/2 ΓC(s) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3375s/2 ΓC(s) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33753375    =    33⋅533^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 1.684341.68434
Root analytic conductor: 1.297821.29782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3375(1376,⋅)\chi_{3375} (1376, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3375, ( :0), 1)(2,\ 3375,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.49346732260.4934673226
L(12)L(\frac12) ≈\approx 0.49346732260.4934673226
L(1)L(1) 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)
bad3 1 1
5 1 1
good2 1+1.82iT−T2 1 + 1.82iT - T^{2}
7 1+T2 1 + T^{2}
11 1−T2 1 - T^{2}
13 1+T2 1 + T^{2}
17 1−1.95iT−T2 1 - 1.95iT - T^{2}
19 1+1.82T+T2 1 + 1.82T + T^{2}
23 1−1.33iT−T2 1 - 1.33iT - T^{2}
29 1−T2 1 - T^{2}
31 1+1.95T+T2 1 + 1.95T + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1−1.61iT−T2 1 - 1.61iT - T^{2}
53 1+0.209iT−T2 1 + 0.209iT - T^{2}
59 1−T2 1 - T^{2}
61 1+0.209T+T2 1 + 0.209T + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−0.209T+T2 1 - 0.209T + T^{2}
83 1+0.209iT−T2 1 + 0.209iT - T^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + 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

−8.955515271445652244572887865592, −8.435397445188503102837135430982, −7.60876447986094351396841774604, −6.34305891947895765050659309041, −5.58742358961995843535992270371, −4.53367544624759941134443907771, −3.89168356983238406897511778764, −3.26461522100164185967484200541, −2.04440955693775002631801303930, −1.56062405406771046491979877722, 0.28790709775048257262888530787, 2.29642077818196458777941923242, 3.65401444200827237478620781999, 4.58733612137639498434243925582, 5.08489111882806646051148188378, 5.95420778777228819897130092559, 6.69214490164254775148805924891, 7.15106380006571344023232963896, 7.906527286251642073995642334849, 8.717264712716054537128525624295

Graph of the ZZ-function along the critical line