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Properties

Label 2-1100-11.5-c1-0-3
Degree 22
Conductor 11001100
Sign −0.781−0.623i-0.781 - 0.623i
Analytic cond. 8.783548.78354
Root an. cond. 2.963702.96370
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  + (1.64 + 1.19i)3-s + (−3.29 + 2.39i)7-s + (0.343 + 1.05i)9-s + (0.00208 + 3.31i)11-s + (−1.80 − 5.56i)13-s + (−2.13 + 6.56i)17-s + (4.84 + 3.51i)19-s − 8.26·21-s − 1.37·23-s + (1.18 − 3.64i)27-s + (−6.12 + 4.44i)29-s + (1.25 + 3.85i)31-s + (−3.94 + 5.44i)33-s + (−7.98 + 5.80i)37-s + (3.66 − 11.2i)39-s + ⋯
L(s)  = 1  + (0.947 + 0.688i)3-s + (−1.24 + 0.905i)7-s + (0.114 + 0.352i)9-s + (0.000630 + 0.999i)11-s + (−0.501 − 1.54i)13-s + (−0.517 + 1.59i)17-s + (1.11 + 0.806i)19-s − 1.80·21-s − 0.287·23-s + (0.227 − 0.700i)27-s + (−1.13 + 0.825i)29-s + (0.225 + 0.693i)31-s + (−0.687 + 0.947i)33-s + (−1.31 + 0.954i)37-s + (0.587 − 1.80i)39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((−0.781−0.623i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+1/2) L(s)=((−0.781−0.623i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: −0.781−0.623i-0.781 - 0.623i
Analytic conductor: 8.783548.78354
Root analytic conductor: 2.963702.96370
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1100(401,⋅)\chi_{1100} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :1/2), −0.781−0.623i)(2,\ 1100,\ (\ :1/2),\ -0.781 - 0.623i)

Particular Values

L(1)L(1) ≈\approx 1.3389745331.338974533
L(12)L(\frac12) ≈\approx 1.3389745331.338974533
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 1
5 1 1
11 1+(−0.00208−3.31i)T 1 + (-0.00208 - 3.31i)T
good3 1+(−1.64−1.19i)T+(0.927+2.85i)T2 1 + (-1.64 - 1.19i)T + (0.927 + 2.85i)T^{2}
7 1+(3.29−2.39i)T+(2.16−6.65i)T2 1 + (3.29 - 2.39i)T + (2.16 - 6.65i)T^{2}
13 1+(1.80+5.56i)T+(−10.5+7.64i)T2 1 + (1.80 + 5.56i)T + (-10.5 + 7.64i)T^{2}
17 1+(2.13−6.56i)T+(−13.7−9.99i)T2 1 + (2.13 - 6.56i)T + (-13.7 - 9.99i)T^{2}
19 1+(−4.84−3.51i)T+(5.87+18.0i)T2 1 + (-4.84 - 3.51i)T + (5.87 + 18.0i)T^{2}
23 1+1.37T+23T2 1 + 1.37T + 23T^{2}
29 1+(6.12−4.44i)T+(8.96−27.5i)T2 1 + (6.12 - 4.44i)T + (8.96 - 27.5i)T^{2}
31 1+(−1.25−3.85i)T+(−25.0+18.2i)T2 1 + (-1.25 - 3.85i)T + (-25.0 + 18.2i)T^{2}
37 1+(7.98−5.80i)T+(11.4−35.1i)T2 1 + (7.98 - 5.80i)T + (11.4 - 35.1i)T^{2}
41 1+(1.79+1.30i)T+(12.6+38.9i)T2 1 + (1.79 + 1.30i)T + (12.6 + 38.9i)T^{2}
43 1+2.49T+43T2 1 + 2.49T + 43T^{2}
47 1+(3.50+2.54i)T+(14.5+44.6i)T2 1 + (3.50 + 2.54i)T + (14.5 + 44.6i)T^{2}
53 1+(−2.83−8.71i)T+(−42.8+31.1i)T2 1 + (-2.83 - 8.71i)T + (-42.8 + 31.1i)T^{2}
59 1+(−7.00+5.09i)T+(18.2−56.1i)T2 1 + (-7.00 + 5.09i)T + (18.2 - 56.1i)T^{2}
61 1+(−1.37+4.24i)T+(−49.3−35.8i)T2 1 + (-1.37 + 4.24i)T + (-49.3 - 35.8i)T^{2}
67 1−2.67T+67T2 1 - 2.67T + 67T^{2}
71 1+(−1.91+5.89i)T+(−57.4−41.7i)T2 1 + (-1.91 + 5.89i)T + (-57.4 - 41.7i)T^{2}
73 1+(−2.80+2.04i)T+(22.5−69.4i)T2 1 + (-2.80 + 2.04i)T + (22.5 - 69.4i)T^{2}
79 1+(−1.24−3.82i)T+(−63.9+46.4i)T2 1 + (-1.24 - 3.82i)T + (-63.9 + 46.4i)T^{2}
83 1+(2.55−7.86i)T+(−67.1−48.7i)T2 1 + (2.55 - 7.86i)T + (-67.1 - 48.7i)T^{2}
89 1−16.9T+89T2 1 - 16.9T + 89T^{2}
97 1+(0.463+1.42i)T+(−78.4+57.0i)T2 1 + (0.463 + 1.42i)T + (-78.4 + 57.0i)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

−10.04657745332939971612554444156, −9.439212401100403283471316407982, −8.635423335338362254126501420583, −7.954932402965466476413861247922, −6.85706946563766720476002201765, −5.86882236207494109491085717529, −5.02901653861688298705586146075, −3.59297774619743882052124664699, −3.23594670269162859809890333702, −2.04441865776561578599086064488, 0.49055501578878481584953446202, 2.17149034660944443814401326299, 3.10096110736859775747037273841, 3.95209372797537468208055550028, 5.21979225152361709584349738477, 6.56530924741580249390146078851, 7.09699473492119564452068954524, 7.66278522706977007284560163238, 8.923191246473913018698864270254, 9.313803066029419150519774517033

Graph of the ZZ-function along the critical line