Properties

Label 2-1100-11.3-c1-0-7
Degree 22
Conductor 11001100
Sign 0.3330.942i0.333 - 0.942i
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  + (0.616 + 1.89i)3-s + (0.139 − 0.430i)7-s + (−0.797 + 0.579i)9-s + (1.01 + 3.15i)11-s + (5.32 − 3.87i)13-s + (−0.959 − 0.696i)17-s + (0.575 + 1.77i)19-s + 0.903·21-s + 5.37·23-s + (3.25 + 2.36i)27-s + (−2.35 + 7.25i)29-s + (−2.83 + 2.05i)31-s + (−5.36 + 3.87i)33-s + (1.78 − 5.50i)37-s + (10.6 + 7.72i)39-s + ⋯
L(s)  = 1  + (0.356 + 1.09i)3-s + (0.0528 − 0.162i)7-s + (−0.265 + 0.193i)9-s + (0.306 + 0.951i)11-s + (1.47 − 1.07i)13-s + (−0.232 − 0.168i)17-s + (0.132 + 0.406i)19-s + 0.197·21-s + 1.11·23-s + (0.626 + 0.454i)27-s + (−0.437 + 1.34i)29-s + (−0.508 + 0.369i)31-s + (−0.934 + 0.675i)33-s + (0.294 − 0.904i)37-s + (1.70 + 1.23i)39-s + ⋯

Functional equation

Λ(s)=(1100s/2ΓC(s)L(s)=((0.3330.942i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1100s/2ΓC(s+1/2)L(s)=((0.3330.942i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    2252112^{2} \cdot 5^{2} \cdot 11
Sign: 0.3330.942i0.333 - 0.942i
Analytic conductor: 8.783548.78354
Root analytic conductor: 2.963702.96370
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1100(201,)\chi_{1100} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :1/2), 0.3330.942i)(2,\ 1100,\ (\ :1/2),\ 0.333 - 0.942i)

Particular Values

L(1)L(1) \approx 2.0274849372.027484937
L(12)L(\frac12) \approx 2.0274849372.027484937
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
11 1+(1.013.15i)T 1 + (-1.01 - 3.15i)T
good3 1+(0.6161.89i)T+(2.42+1.76i)T2 1 + (-0.616 - 1.89i)T + (-2.42 + 1.76i)T^{2}
7 1+(0.139+0.430i)T+(5.664.11i)T2 1 + (-0.139 + 0.430i)T + (-5.66 - 4.11i)T^{2}
13 1+(5.32+3.87i)T+(4.0112.3i)T2 1 + (-5.32 + 3.87i)T + (4.01 - 12.3i)T^{2}
17 1+(0.959+0.696i)T+(5.25+16.1i)T2 1 + (0.959 + 0.696i)T + (5.25 + 16.1i)T^{2}
19 1+(0.5751.77i)T+(15.3+11.1i)T2 1 + (-0.575 - 1.77i)T + (-15.3 + 11.1i)T^{2}
23 15.37T+23T2 1 - 5.37T + 23T^{2}
29 1+(2.357.25i)T+(23.417.0i)T2 1 + (2.35 - 7.25i)T + (-23.4 - 17.0i)T^{2}
31 1+(2.832.05i)T+(9.5729.4i)T2 1 + (2.83 - 2.05i)T + (9.57 - 29.4i)T^{2}
37 1+(1.78+5.50i)T+(29.921.7i)T2 1 + (-1.78 + 5.50i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.1030.317i)T+(33.1+24.0i)T2 1 + (-0.103 - 0.317i)T + (-33.1 + 24.0i)T^{2}
43 1+10.9T+43T2 1 + 10.9T + 43T^{2}
47 1+(1.986.09i)T+(38.0+27.6i)T2 1 + (-1.98 - 6.09i)T + (-38.0 + 27.6i)T^{2}
53 1+(1.85+1.34i)T+(16.350.4i)T2 1 + (-1.85 + 1.34i)T + (16.3 - 50.4i)T^{2}
59 1+(0.449+1.38i)T+(47.734.6i)T2 1 + (-0.449 + 1.38i)T + (-47.7 - 34.6i)T^{2}
61 1+(7.22+5.25i)T+(18.8+58.0i)T2 1 + (7.22 + 5.25i)T + (18.8 + 58.0i)T^{2}
67 1+4.37T+67T2 1 + 4.37T + 67T^{2}
71 1+(8.916.47i)T+(21.9+67.5i)T2 1 + (-8.91 - 6.47i)T + (21.9 + 67.5i)T^{2}
73 1+(1.09+3.36i)T+(59.042.9i)T2 1 + (-1.09 + 3.36i)T + (-59.0 - 42.9i)T^{2}
79 1+(2.46+1.79i)T+(24.475.1i)T2 1 + (-2.46 + 1.79i)T + (24.4 - 75.1i)T^{2}
83 1+(7.95+5.78i)T+(25.6+78.9i)T2 1 + (7.95 + 5.78i)T + (25.6 + 78.9i)T^{2}
89 1+12.7T+89T2 1 + 12.7T + 89T^{2}
97 1+(11.2+8.14i)T+(29.992.2i)T2 1 + (-11.2 + 8.14i)T + (29.9 - 92.2i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.03618256127712093917466993124, −9.160259433824650837182874010031, −8.678504594239662058413322518028, −7.58505125601291003457702591486, −6.71176678924710145473234029711, −5.55590767046173357915791620526, −4.72292847622825239641666668641, −3.77094475484248375579283793374, −3.10733758403803069709267989494, −1.39739620367811548993509045386, 1.03548185214962837627981561849, 2.05948770301400677078185571941, 3.28937156521283294639204650881, 4.32136561355290715626797695441, 5.68673039256893310767262011345, 6.51070959804564568603951208413, 7.06812684004826972520958144585, 8.198100081661643437888171141825, 8.651394545916483888951272032762, 9.434270840305318780711532454578

Graph of the ZZ-function along the critical line