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Properties

Label 2-550-11.4-c1-0-11
Degree 22
Conductor 550550
Sign 0.530+0.847i0.530 + 0.847i
Analytic cond. 4.391774.39177
Root an. cond. 2.095652.09565
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  + (−0.809 − 0.587i)2-s + (0.381 − 1.17i)3-s + (0.309 + 0.951i)4-s + (−1 + 0.726i)6-s + (−0.190 − 0.587i)7-s + (0.309 − 0.951i)8-s + (1.19 + 0.865i)9-s + (1.69 + 2.85i)11-s + 1.23·12-s + (1.92 + 1.40i)13-s + (−0.190 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (5.23 − 3.80i)17-s + (−0.454 − 1.40i)18-s + (1.11 − 3.44i)19-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.220 − 0.678i)3-s + (0.154 + 0.475i)4-s + (−0.408 + 0.296i)6-s + (−0.0721 − 0.222i)7-s + (0.109 − 0.336i)8-s + (0.396 + 0.288i)9-s + (0.509 + 0.860i)11-s + 0.356·12-s + (0.534 + 0.388i)13-s + (−0.0510 + 0.157i)14-s + (−0.202 + 0.146i)16-s + (1.26 − 0.922i)17-s + (−0.107 − 0.330i)18-s + (0.256 − 0.789i)19-s + ⋯

Functional equation

Λ(s)=(550s/2 ΓC(s) L(s)=((0.530+0.847i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(550s/2 ΓC(s+1/2) L(s)=((0.530+0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 550550    =    2⋅52⋅112 \cdot 5^{2} \cdot 11
Sign: 0.530+0.847i0.530 + 0.847i
Analytic conductor: 4.391774.39177
Root analytic conductor: 2.095652.09565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ550(301,⋅)\chi_{550} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 550, ( :1/2), 0.530+0.847i)(2,\ 550,\ (\ :1/2),\ 0.530 + 0.847i)

Particular Values

L(1)L(1) ≈\approx 1.14089−0.632202i1.14089 - 0.632202i
L(12)L(\frac12) ≈\approx 1.14089−0.632202i1.14089 - 0.632202i
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+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
5 1 1
11 1+(−1.69−2.85i)T 1 + (-1.69 - 2.85i)T
good3 1+(−0.381+1.17i)T+(−2.42−1.76i)T2 1 + (-0.381 + 1.17i)T + (-2.42 - 1.76i)T^{2}
7 1+(0.190+0.587i)T+(−5.66+4.11i)T2 1 + (0.190 + 0.587i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.92−1.40i)T+(4.01+12.3i)T2 1 + (-1.92 - 1.40i)T + (4.01 + 12.3i)T^{2}
17 1+(−5.23+3.80i)T+(5.25−16.1i)T2 1 + (-5.23 + 3.80i)T + (5.25 - 16.1i)T^{2}
19 1+(−1.11+3.44i)T+(−15.3−11.1i)T2 1 + (-1.11 + 3.44i)T + (-15.3 - 11.1i)T^{2}
23 1+4.61T+23T2 1 + 4.61T + 23T^{2}
29 1+(−1.38−4.25i)T+(−23.4+17.0i)T2 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2}
31 1+(1.61+1.17i)T+(9.57+29.4i)T2 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2}
37 1+(1.73+5.34i)T+(−29.9+21.7i)T2 1 + (1.73 + 5.34i)T + (-29.9 + 21.7i)T^{2}
41 1+(−0.190+0.587i)T+(−33.1−24.0i)T2 1 + (-0.190 + 0.587i)T + (-33.1 - 24.0i)T^{2}
43 1−8.47T+43T2 1 - 8.47T + 43T^{2}
47 1+(3.11−9.59i)T+(−38.0−27.6i)T2 1 + (3.11 - 9.59i)T + (-38.0 - 27.6i)T^{2}
53 1+(5.73+4.16i)T+(16.3+50.4i)T2 1 + (5.73 + 4.16i)T + (16.3 + 50.4i)T^{2}
59 1+(3.35+10.3i)T+(−47.7+34.6i)T2 1 + (3.35 + 10.3i)T + (-47.7 + 34.6i)T^{2}
61 1+(−5.61+4.08i)T+(18.8−58.0i)T2 1 + (-5.61 + 4.08i)T + (18.8 - 58.0i)T^{2}
67 1+9.23T+67T2 1 + 9.23T + 67T^{2}
71 1+(1.61−1.17i)T+(21.9−67.5i)T2 1 + (1.61 - 1.17i)T + (21.9 - 67.5i)T^{2}
73 1+(0.472+1.45i)T+(−59.0+42.9i)T2 1 + (0.472 + 1.45i)T + (-59.0 + 42.9i)T^{2}
79 1+(−5−3.63i)T+(24.4+75.1i)T2 1 + (-5 - 3.63i)T + (24.4 + 75.1i)T^{2}
83 1+(8.23−5.98i)T+(25.6−78.9i)T2 1 + (8.23 - 5.98i)T + (25.6 - 78.9i)T^{2}
89 1−11.3T+89T2 1 - 11.3T + 89T^{2}
97 1+(−7.47−5.42i)T+(29.9+92.2i)T2 1 + (-7.47 - 5.42i)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

−10.57583586837593087098802350857, −9.702259361282543488058948923074, −9.032959904270540800332098109175, −7.79886947143502242021820552452, −7.32816312943591065330930594856, −6.41679134912578756343899467374, −4.89562921503912325200188911230, −3.69215664260004750754559127335, −2.30802140684190887493036530957, −1.16326285457883077596620939980, 1.30428425196737489946147590330, 3.27442215457065391043792652179, 4.15022025792735909581657036227, 5.68956763439249199250382304885, 6.20758023619261611541975553191, 7.57066559309663092735657629839, 8.391007945990857418779274943456, 9.117188317937824126051645099847, 10.11983851575812512832974468460, 10.45488803271760838444948290234

Graph of the ZZ-function along the critical line