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Properties

Label 2-1100-5.4-c5-0-10
Degree 22
Conductor 11001100
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 176.422176.422
Root an. cond. 13.282413.2824
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.426i·3-s − 59.7i·7-s + 242.·9-s − 121·11-s + 1.12e3i·13-s + 1.94e3i·17-s + 1.24e3·19-s + 25.5·21-s + 4.64e3i·23-s + 207. i·27-s − 1.00e3·29-s − 2.47e3·31-s − 51.6i·33-s − 1.08e4i·37-s − 480.·39-s + ⋯
L(s)  = 1  + 0.0273i·3-s − 0.461i·7-s + 0.999·9-s − 0.301·11-s + 1.84i·13-s + 1.63i·17-s + 0.791·19-s + 0.0126·21-s + 1.82i·23-s + 0.0547i·27-s − 0.220·29-s − 0.463·31-s − 0.00825i·33-s − 1.30i·37-s − 0.0505·39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+5/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 176.422176.422
Root analytic conductor: 13.282413.2824
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ1100(749,⋅)\chi_{1100} (749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :5/2), −0.894−0.447i)(2,\ 1100,\ (\ :5/2),\ -0.894 - 0.447i)

Particular Values

L(3)L(3) ≈\approx 1.2790841231.279084123
L(12)L(\frac12) ≈\approx 1.2790841231.279084123
L(72)L(\frac{7}{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
5 1 1
11 1+121T 1 + 121T
good3 1−0.426iT−243T2 1 - 0.426iT - 243T^{2}
7 1+59.7iT−1.68e4T2 1 + 59.7iT - 1.68e4T^{2}
13 1−1.12e3iT−3.71e5T2 1 - 1.12e3iT - 3.71e5T^{2}
17 1−1.94e3iT−1.41e6T2 1 - 1.94e3iT - 1.41e6T^{2}
19 1−1.24e3T+2.47e6T2 1 - 1.24e3T + 2.47e6T^{2}
23 1−4.64e3iT−6.43e6T2 1 - 4.64e3iT - 6.43e6T^{2}
29 1+1.00e3T+2.05e7T2 1 + 1.00e3T + 2.05e7T^{2}
31 1+2.47e3T+2.86e7T2 1 + 2.47e3T + 2.86e7T^{2}
37 1+1.08e4iT−6.93e7T2 1 + 1.08e4iT - 6.93e7T^{2}
41 1+1.17e4T+1.15e8T2 1 + 1.17e4T + 1.15e8T^{2}
43 1+6.90e3iT−1.47e8T2 1 + 6.90e3iT - 1.47e8T^{2}
47 1+4.95e3iT−2.29e8T2 1 + 4.95e3iT - 2.29e8T^{2}
53 1+8.17e3iT−4.18e8T2 1 + 8.17e3iT - 4.18e8T^{2}
59 1+3.69e4T+7.14e8T2 1 + 3.69e4T + 7.14e8T^{2}
61 1+1.18e4T+8.44e8T2 1 + 1.18e4T + 8.44e8T^{2}
67 1+825.iT−1.35e9T2 1 + 825. iT - 1.35e9T^{2}
71 1+1.19e4T+1.80e9T2 1 + 1.19e4T + 1.80e9T^{2}
73 1−5.90e4iT−2.07e9T2 1 - 5.90e4iT - 2.07e9T^{2}
79 1−2.63e3T+3.07e9T2 1 - 2.63e3T + 3.07e9T^{2}
83 1+1.08e5iT−3.93e9T2 1 + 1.08e5iT - 3.93e9T^{2}
89 1+3.21e4T+5.58e9T2 1 + 3.21e4T + 5.58e9T^{2}
97 1−1.65e5iT−8.58e9T2 1 - 1.65e5iT - 8.58e9T^{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

−9.498560079866016655505892392567, −8.791865829040937068101513601527, −7.57754385064486987523433848733, −7.15933733385467007688169452934, −6.19187474705967285284262471136, −5.17732855128147697143688378791, −4.07703658589769479182515312991, −3.64194948117482996007648771563, −1.90963578624285681922389540821, −1.38745606187390969933632660655, 0.24146969983669057302087796865, 1.15789956436578743935704241579, 2.58220651382054327378960409091, 3.22412698211312693734720984964, 4.65559650281075897363165264012, 5.21932350151712180157554188815, 6.24445235678337608286356794733, 7.24790683299201263957619928648, 7.85971102231806112246870598109, 8.762323890089930716166785355517

Graph of the ZZ-function along the critical line