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Properties

Label 2-48e2-4.3-c2-0-67
Degree 22
Conductor 23042304
Sign −1-1
Analytic cond. 62.779462.7794
Root an. cond. 7.923347.92334
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 9.79i·7-s + 19.5i·11-s − 21·25-s + 50·29-s − 48.9i·31-s + 19.5i·35-s − 46.9·49-s − 94·53-s − 39.1i·55-s − 117. i·59-s − 50·73-s + 191.·77-s + 146. i·79-s − 97.9i·83-s + ⋯
L(s)  = 1  − 0.400·5-s − 1.39i·7-s + 1.78i·11-s − 0.839·25-s + 1.72·29-s − 1.58i·31-s + 0.559i·35-s − 0.959·49-s − 1.77·53-s − 0.712i·55-s − 1.99i·59-s − 0.684·73-s + 2.49·77-s + 1.86i·79-s − 1.18i·83-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s) L(s)=(− Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+1) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: −1-1
Analytic conductor: 62.779462.7794
Root analytic conductor: 7.923347.92334
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2304(1279,⋅)\chi_{2304} (1279, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2304, ( :1), −1)(2,\ 2304,\ (\ :1),\ -1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.22299469840.2229946984
L(12)L(\frac12) ≈\approx 0.22299469840.2229946984
L(2)L(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
3 1 1
good5 1+2T+25T2 1 + 2T + 25T^{2}
7 1+9.79iT−49T2 1 + 9.79iT - 49T^{2}
11 1−19.5iT−121T2 1 - 19.5iT - 121T^{2}
13 1+169T2 1 + 169T^{2}
17 1+289T2 1 + 289T^{2}
19 1−361T2 1 - 361T^{2}
23 1−529T2 1 - 529T^{2}
29 1−50T+841T2 1 - 50T + 841T^{2}
31 1+48.9iT−961T2 1 + 48.9iT - 961T^{2}
37 1+1.36e3T2 1 + 1.36e3T^{2}
41 1+1.68e3T2 1 + 1.68e3T^{2}
43 1−1.84e3T2 1 - 1.84e3T^{2}
47 1−2.20e3T2 1 - 2.20e3T^{2}
53 1+94T+2.80e3T2 1 + 94T + 2.80e3T^{2}
59 1+117.iT−3.48e3T2 1 + 117. iT - 3.48e3T^{2}
61 1+3.72e3T2 1 + 3.72e3T^{2}
67 1−4.48e3T2 1 - 4.48e3T^{2}
71 1−5.04e3T2 1 - 5.04e3T^{2}
73 1+50T+5.32e3T2 1 + 50T + 5.32e3T^{2}
79 1−146.iT−6.24e3T2 1 - 146. iT - 6.24e3T^{2}
83 1+97.9iT−6.88e3T2 1 + 97.9iT - 6.88e3T^{2}
89 1+7.92e3T2 1 + 7.92e3T^{2}
97 1+190T+9.40e3T2 1 + 190T + 9.40e3T^{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

−8.126977724082118074478874345826, −7.70263637832272259513588172499, −6.97251431510453506260068885846, −6.34395569068220927675544227725, −4.98565696473563310666780974391, −4.34798836266470013911298588525, −3.74754893144402749709364964754, −2.43995065001743789552614979359, −1.31733389704277170242792104640, −0.05661955552560939533794860658, 1.33400402215906214233691604269, 2.74696020645240628597123045673, 3.24359822542426417566104448901, 4.44004974852518002104619470631, 5.45018670817730035360500465058, 5.98103537922693228558033589354, 6.75661426319547008065893387123, 7.933623973047014331030603543449, 8.551321426911397041503909628457, 8.902445189812925091754232267700

Graph of the ZZ-function along the critical line