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Properties

Label 2-1100-11.4-c1-0-17
Degree 22
Conductor 11001100
Sign −0.987+0.156i-0.987 + 0.156i
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  + (0.954 − 2.93i)3-s + (−0.787 − 2.42i)7-s + (−5.29 − 3.85i)9-s + (2.91 + 1.57i)11-s + (−3.13 − 2.27i)13-s + (5.39 − 3.92i)17-s + (−0.918 + 2.82i)19-s − 7.87·21-s − 2.50·23-s + (−8.87 + 6.44i)27-s + (−1.00 − 3.08i)29-s + (−3.01 − 2.18i)31-s + (7.41 − 7.07i)33-s + (−0.630 − 1.94i)37-s + (−9.67 + 7.02i)39-s + ⋯
L(s)  = 1  + (0.551 − 1.69i)3-s + (−0.297 − 0.916i)7-s + (−1.76 − 1.28i)9-s + (0.880 + 0.474i)11-s + (−0.868 − 0.630i)13-s + (1.30 − 0.951i)17-s + (−0.210 + 0.648i)19-s − 1.71·21-s − 0.521·23-s + (−1.70 + 1.24i)27-s + (−0.185 − 0.571i)29-s + (−0.541 − 0.393i)31-s + (1.29 − 1.23i)33-s + (−0.103 − 0.319i)37-s + (−1.54 + 1.12i)39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((−0.987+0.156i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+1/2) L(s)=((−0.987+0.156i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: −0.987+0.156i-0.987 + 0.156i
Analytic conductor: 8.783548.78354
Root analytic conductor: 2.963702.96370
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1100(301,⋅)\chi_{1100} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :1/2), −0.987+0.156i)(2,\ 1100,\ (\ :1/2),\ -0.987 + 0.156i)

Particular Values

L(1)L(1) ≈\approx 1.5871300571.587130057
L(12)L(\frac12) ≈\approx 1.5871300571.587130057
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+(−2.91−1.57i)T 1 + (-2.91 - 1.57i)T
good3 1+(−0.954+2.93i)T+(−2.42−1.76i)T2 1 + (-0.954 + 2.93i)T + (-2.42 - 1.76i)T^{2}
7 1+(0.787+2.42i)T+(−5.66+4.11i)T2 1 + (0.787 + 2.42i)T + (-5.66 + 4.11i)T^{2}
13 1+(3.13+2.27i)T+(4.01+12.3i)T2 1 + (3.13 + 2.27i)T + (4.01 + 12.3i)T^{2}
17 1+(−5.39+3.92i)T+(5.25−16.1i)T2 1 + (-5.39 + 3.92i)T + (5.25 - 16.1i)T^{2}
19 1+(0.918−2.82i)T+(−15.3−11.1i)T2 1 + (0.918 - 2.82i)T + (-15.3 - 11.1i)T^{2}
23 1+2.50T+23T2 1 + 2.50T + 23T^{2}
29 1+(1.00+3.08i)T+(−23.4+17.0i)T2 1 + (1.00 + 3.08i)T + (-23.4 + 17.0i)T^{2}
31 1+(3.01+2.18i)T+(9.57+29.4i)T2 1 + (3.01 + 2.18i)T + (9.57 + 29.4i)T^{2}
37 1+(0.630+1.94i)T+(−29.9+21.7i)T2 1 + (0.630 + 1.94i)T + (-29.9 + 21.7i)T^{2}
41 1+(3.15−9.70i)T+(−33.1−24.0i)T2 1 + (3.15 - 9.70i)T + (-33.1 - 24.0i)T^{2}
43 1+6.30T+43T2 1 + 6.30T + 43T^{2}
47 1+(0.894−2.75i)T+(−38.0−27.6i)T2 1 + (0.894 - 2.75i)T + (-38.0 - 27.6i)T^{2}
53 1+(−7.36−5.35i)T+(16.3+50.4i)T2 1 + (-7.36 - 5.35i)T + (16.3 + 50.4i)T^{2}
59 1+(2.44+7.52i)T+(−47.7+34.6i)T2 1 + (2.44 + 7.52i)T + (-47.7 + 34.6i)T^{2}
61 1+(−5.03+3.65i)T+(18.8−58.0i)T2 1 + (-5.03 + 3.65i)T + (18.8 - 58.0i)T^{2}
67 1−7.40T+67T2 1 - 7.40T + 67T^{2}
71 1+(−5.48+3.98i)T+(21.9−67.5i)T2 1 + (-5.48 + 3.98i)T + (21.9 - 67.5i)T^{2}
73 1+(−4.28−13.1i)T+(−59.0+42.9i)T2 1 + (-4.28 - 13.1i)T + (-59.0 + 42.9i)T^{2}
79 1+(−2.17−1.57i)T+(24.4+75.1i)T2 1 + (-2.17 - 1.57i)T + (24.4 + 75.1i)T^{2}
83 1+(−2.08+1.51i)T+(25.6−78.9i)T2 1 + (-2.08 + 1.51i)T + (25.6 - 78.9i)T^{2}
89 1−2.81T+89T2 1 - 2.81T + 89T^{2}
97 1+(5.31+3.85i)T+(29.9+92.2i)T2 1 + (5.31 + 3.85i)T + (29.9 + 92.2i)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

−9.574703445641296651314989779888, −8.265189751690257269964055731632, −7.70030053797639874606434592799, −7.10173174228256397135697120567, −6.42787007125683610233051832749, −5.37184577149157435594411052852, −3.89192705115544463894265134191, −2.92048416757130542480752107005, −1.75573697656360834220703187259, −0.64921350695099055962329605544, 2.17491140341382675264651152013, 3.35246615918605315756715644897, 3.92911803529490097884212909734, 5.06637385098441315245197436912, 5.68726348424750505210417909062, 6.84399854214729492952180096294, 8.189842135971418573514465817143, 8.878840203466239108361000426536, 9.357552634994354402188624761357, 10.11152427532529935858780139844

Graph of the ZZ-function along the critical line