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Properties

Label 2-575-23.22-c2-0-44
Degree 22
Conductor 575575
Sign ii
Analytic cond. 15.667615.6676
Root an. cond. 3.958233.95823
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  − 4·4-s + 9i·7-s − 9·9-s + 16·16-s − 11i·17-s − 23i·23-s − 36i·28-s + 57·29-s − 53·31-s + 36·36-s − 51i·37-s − 33·41-s − 6i·43-s − 32·49-s − 101i·53-s + ⋯
L(s)  = 1  − 4-s + 1.28i·7-s − 9-s + 16-s − 0.647i·17-s − i·23-s − 1.28i·28-s + 1.96·29-s − 1.70·31-s + 36-s − 1.37i·37-s − 0.804·41-s − 0.139i·43-s − 0.653·49-s − 1.90i·53-s + ⋯

Functional equation

Λ(s)=(575s/2 ΓC(s) L(s)=(i Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(575s/2 ΓC(s+1) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 575575    =    52⋅235^{2} \cdot 23
Sign: ii
Analytic conductor: 15.667615.6676
Root analytic conductor: 3.958233.95823
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ575(551,⋅)\chi_{575} (551, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 575, ( :1), i)(2,\ 575,\ (\ :1),\ i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.62781495820.6278149582
L(12)L(\frac12) ≈\approx 0.62781495820.6278149582
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)
bad5 1 1
23 1+23iT 1 + 23iT
good2 1+4T2 1 + 4T^{2}
3 1+9T2 1 + 9T^{2}
7 1−9iT−49T2 1 - 9iT - 49T^{2}
11 1−121T2 1 - 121T^{2}
13 1+169T2 1 + 169T^{2}
17 1+11iT−289T2 1 + 11iT - 289T^{2}
19 1−361T2 1 - 361T^{2}
29 1−57T+841T2 1 - 57T + 841T^{2}
31 1+53T+961T2 1 + 53T + 961T^{2}
37 1+51iT−1.36e3T2 1 + 51iT - 1.36e3T^{2}
41 1+33T+1.68e3T2 1 + 33T + 1.68e3T^{2}
43 1+6iT−1.84e3T2 1 + 6iT - 1.84e3T^{2}
47 1+2.20e3T2 1 + 2.20e3T^{2}
53 1+101iT−2.80e3T2 1 + 101iT - 2.80e3T^{2}
59 1+3T+3.48e3T2 1 + 3T + 3.48e3T^{2}
61 1−3.72e3T2 1 - 3.72e3T^{2}
67 1+111iT−4.48e3T2 1 + 111iT - 4.48e3T^{2}
71 1−27T+5.04e3T2 1 - 27T + 5.04e3T^{2}
73 1+5.32e3T2 1 + 5.32e3T^{2}
79 1−6.24e3T2 1 - 6.24e3T^{2}
83 1+41iT−6.88e3T2 1 + 41iT - 6.88e3T^{2}
89 1−7.92e3T2 1 - 7.92e3T^{2}
97 1−174iT−9.40e3T2 1 - 174iT - 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

−10.18639044708063414484515009731, −9.112746519857093525942293249269, −8.770644637692087626720310551335, −7.944348273142579988588109488481, −6.48787221812149731211365780830, −5.49290367521561014015759795903, −4.89131446991755701886211113823, −3.46152165171184711003592216770, −2.35155000208740894444951165322, −0.28379869387443741491947971854, 1.14843055427792614031553325437, 3.18167906418851621428924614159, 4.09429201901764397941036095313, 5.07282144531352294338782397822, 6.09654011047177270819405022452, 7.29017626822810836817548691053, 8.205352300985606674228640565642, 8.902425802114176384682052524904, 9.948904697411811935954663266421, 10.56958895106232392908395374872

Graph of the ZZ-function along the critical line