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Properties

Label 2-1100-11.3-c1-0-16
Degree 22
Conductor 11001100
Sign 0.204+0.978i0.204 + 0.978i
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  + (0.321 + 0.989i)3-s + (0.699 − 2.15i)7-s + (1.55 − 1.12i)9-s + (−2.50 − 2.16i)11-s + (−0.217 + 0.157i)13-s + (−4.79 − 3.48i)17-s + (−1.77 − 5.46i)19-s + 2.35·21-s − 2.92·23-s + (4.13 + 3.00i)27-s + (2.00 − 6.16i)29-s + (−0.202 + 0.146i)31-s + (1.34 − 3.17i)33-s + (−3.18 + 9.79i)37-s + (−0.225 − 0.164i)39-s + ⋯
L(s)  = 1  + (0.185 + 0.571i)3-s + (0.264 − 0.813i)7-s + (0.517 − 0.375i)9-s + (−0.756 − 0.654i)11-s + (−0.0601 + 0.0437i)13-s + (−1.16 − 0.844i)17-s + (−0.407 − 1.25i)19-s + 0.513·21-s − 0.610·23-s + (0.796 + 0.578i)27-s + (0.371 − 1.14i)29-s + (−0.0363 + 0.0263i)31-s + (0.233 − 0.553i)33-s + (−0.523 + 1.61i)37-s + (−0.0361 − 0.0262i)39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((0.204+0.978i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+1/2) L(s)=((0.204+0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 0.204+0.978i0.204 + 0.978i
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), 0.204+0.978i)(2,\ 1100,\ (\ :1/2),\ 0.204 + 0.978i)

Particular Values

L(1)L(1) ≈\approx 1.3490203031.349020303
L(12)L(\frac12) ≈\approx 1.3490203031.349020303
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+(2.50+2.16i)T 1 + (2.50 + 2.16i)T
good3 1+(−0.321−0.989i)T+(−2.42+1.76i)T2 1 + (-0.321 - 0.989i)T + (-2.42 + 1.76i)T^{2}
7 1+(−0.699+2.15i)T+(−5.66−4.11i)T2 1 + (-0.699 + 2.15i)T + (-5.66 - 4.11i)T^{2}
13 1+(0.217−0.157i)T+(4.01−12.3i)T2 1 + (0.217 - 0.157i)T + (4.01 - 12.3i)T^{2}
17 1+(4.79+3.48i)T+(5.25+16.1i)T2 1 + (4.79 + 3.48i)T + (5.25 + 16.1i)T^{2}
19 1+(1.77+5.46i)T+(−15.3+11.1i)T2 1 + (1.77 + 5.46i)T + (-15.3 + 11.1i)T^{2}
23 1+2.92T+23T2 1 + 2.92T + 23T^{2}
29 1+(−2.00+6.16i)T+(−23.4−17.0i)T2 1 + (-2.00 + 6.16i)T + (-23.4 - 17.0i)T^{2}
31 1+(0.202−0.146i)T+(9.57−29.4i)T2 1 + (0.202 - 0.146i)T + (9.57 - 29.4i)T^{2}
37 1+(3.18−9.79i)T+(−29.9−21.7i)T2 1 + (3.18 - 9.79i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.730+2.24i)T+(−33.1+24.0i)T2 1 + (0.730 + 2.24i)T + (-33.1 + 24.0i)T^{2}
43 1+7.56T+43T2 1 + 7.56T + 43T^{2}
47 1+(0.349+1.07i)T+(−38.0+27.6i)T2 1 + (0.349 + 1.07i)T + (-38.0 + 27.6i)T^{2}
53 1+(−8.25+5.99i)T+(16.3−50.4i)T2 1 + (-8.25 + 5.99i)T + (16.3 - 50.4i)T^{2}
59 1+(−3.52+10.8i)T+(−47.7−34.6i)T2 1 + (-3.52 + 10.8i)T + (-47.7 - 34.6i)T^{2}
61 1+(−3.73−2.71i)T+(18.8+58.0i)T2 1 + (-3.73 - 2.71i)T + (18.8 + 58.0i)T^{2}
67 1−6.11T+67T2 1 - 6.11T + 67T^{2}
71 1+(4.11+2.99i)T+(21.9+67.5i)T2 1 + (4.11 + 2.99i)T + (21.9 + 67.5i)T^{2}
73 1+(2.52−7.75i)T+(−59.0−42.9i)T2 1 + (2.52 - 7.75i)T + (-59.0 - 42.9i)T^{2}
79 1+(−6.71+4.88i)T+(24.4−75.1i)T2 1 + (-6.71 + 4.88i)T + (24.4 - 75.1i)T^{2}
83 1+(−13.2−9.65i)T+(25.6+78.9i)T2 1 + (-13.2 - 9.65i)T + (25.6 + 78.9i)T^{2}
89 1+5.73T+89T2 1 + 5.73T + 89T^{2}
97 1+(−3.65+2.65i)T+(29.9−92.2i)T2 1 + (-3.65 + 2.65i)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

−9.828381758482993571199217041343, −8.850116477921805230217408626441, −8.166978346734256910933914153489, −7.07609766006653730819255393288, −6.50640285503763083519348132425, −5.07337401372084203913481600088, −4.47223481188656011912914839372, −3.51201877451047481420628366329, −2.34854775999596368887724224267, −0.56235989430303020513116724707, 1.77722994618713293408940274423, 2.35614257417165989008524587201, 3.89943946449292961670933999378, 4.90566058467562133030035121648, 5.82110556296640175344503208846, 6.77679008099699964515567885934, 7.61576379018627170302788119060, 8.342342903252350967602849441752, 9.001831118150046083668826570749, 10.28451455486558671019876222447

Graph of the ZZ-function along the critical line