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Properties

Label 2-1100-55.54-c2-0-28
Degree 22
Conductor 11001100
Sign 0.850−0.525i0.850 - 0.525i
Analytic cond. 29.972829.9728
Root an. cond. 5.474745.47474
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  + 4.69i·3-s + 7.54·7-s − 13.0·9-s + (5.78 − 9.35i)11-s − 0.829·13-s + 29.8·17-s − 36.2i·19-s + 35.4i·21-s − 29.7i·23-s − 18.9i·27-s + 10.0i·29-s + 34.6·31-s + (43.9 + 27.1i)33-s − 61.9i·37-s − 3.89i·39-s + ⋯
L(s)  = 1  + 1.56i·3-s + 1.07·7-s − 1.44·9-s + (0.525 − 0.850i)11-s − 0.0637·13-s + 1.75·17-s − 1.90i·19-s + 1.68i·21-s − 1.29i·23-s − 0.700i·27-s + 0.346i·29-s + 1.11·31-s + (1.33 + 0.822i)33-s − 1.67i·37-s − 0.0997i·39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((0.850−0.525i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+1) L(s)=((0.850−0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 0.850−0.525i0.850 - 0.525i
Analytic conductor: 29.972829.9728
Root analytic conductor: 5.474745.47474
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1100(549,⋅)\chi_{1100} (549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :1), 0.850−0.525i)(2,\ 1100,\ (\ :1),\ 0.850 - 0.525i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.3429395212.342939521
L(12)L(\frac12) ≈\approx 2.3429395212.342939521
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
5 1 1
11 1+(−5.78+9.35i)T 1 + (-5.78 + 9.35i)T
good3 1−4.69iT−9T2 1 - 4.69iT - 9T^{2}
7 1−7.54T+49T2 1 - 7.54T + 49T^{2}
13 1+0.829T+169T2 1 + 0.829T + 169T^{2}
17 1−29.8T+289T2 1 - 29.8T + 289T^{2}
19 1+36.2iT−361T2 1 + 36.2iT - 361T^{2}
23 1+29.7iT−529T2 1 + 29.7iT - 529T^{2}
29 1−10.0iT−841T2 1 - 10.0iT - 841T^{2}
31 1−34.6T+961T2 1 - 34.6T + 961T^{2}
37 1+61.9iT−1.36e3T2 1 + 61.9iT - 1.36e3T^{2}
41 1+11.1iT−1.68e3T2 1 + 11.1iT - 1.68e3T^{2}
43 1−39.4T+1.84e3T2 1 - 39.4T + 1.84e3T^{2}
47 1−6.35iT−2.20e3T2 1 - 6.35iT - 2.20e3T^{2}
53 1+56.5iT−2.80e3T2 1 + 56.5iT - 2.80e3T^{2}
59 1+70.4T+3.48e3T2 1 + 70.4T + 3.48e3T^{2}
61 1+8.41iT−3.72e3T2 1 + 8.41iT - 3.72e3T^{2}
67 1+18.7iT−4.48e3T2 1 + 18.7iT - 4.48e3T^{2}
71 1+3.79T+5.04e3T2 1 + 3.79T + 5.04e3T^{2}
73 1+70.9T+5.32e3T2 1 + 70.9T + 5.32e3T^{2}
79 1−127.iT−6.24e3T2 1 - 127. iT - 6.24e3T^{2}
83 1+100.T+6.88e3T2 1 + 100.T + 6.88e3T^{2}
89 1−66.0T+7.92e3T2 1 - 66.0T + 7.92e3T^{2}
97 1+1.65iT−9.40e3T2 1 + 1.65iT - 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

−9.742751728313078323390183695516, −8.954087275229390455807690756914, −8.406403965645817884524761511712, −7.37010976257593535298229262434, −6.09203069853569857812647222221, −5.15718883552806384938603911286, −4.59266667857322881501224142278, −3.65584104114005084935008971426, −2.66373460349110374386882008648, −0.829515958874335624927622064239, 1.36276272872793424305800197059, 1.57393253202348613038079479609, 3.06689935974261009279633362357, 4.39041434988422098757748280470, 5.58325401914092580644596854042, 6.22921966571941195336078055329, 7.38955850320620024258800676602, 7.77677622563457311914279031628, 8.332105731958723532226905172875, 9.634211636541128893106978548597

Graph of the ZZ-function along the critical line