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Properties

Label 1-1053-1053.119-r0-0-0
Degree 11
Conductor 10531053
Sign 0.923−0.384i0.923 - 0.384i
Analytic cond. 4.890114.89011
Root an. cond. 4.890114.89011
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.998 − 0.0581i)2-s + (0.993 − 0.116i)4-s + (−0.727 − 0.686i)5-s + (−0.918 + 0.396i)7-s + (0.984 − 0.173i)8-s + (−0.766 − 0.642i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)14-s + (0.973 − 0.230i)16-s + (0.766 + 0.642i)17-s + (0.984 − 0.173i)19-s + (−0.802 − 0.597i)20-s + (−0.686 + 0.727i)22-s + (0.396 − 0.918i)23-s + (0.0581 + 0.998i)25-s + ⋯
L(s)  = 1  + (0.998 − 0.0581i)2-s + (0.993 − 0.116i)4-s + (−0.727 − 0.686i)5-s + (−0.918 + 0.396i)7-s + (0.984 − 0.173i)8-s + (−0.766 − 0.642i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)14-s + (0.973 − 0.230i)16-s + (0.766 + 0.642i)17-s + (0.984 − 0.173i)19-s + (−0.802 − 0.597i)20-s + (−0.686 + 0.727i)22-s + (0.396 − 0.918i)23-s + (0.0581 + 0.998i)25-s + ⋯

Functional equation

Λ(s)=(1053s/2 ΓR(s) L(s)=((0.923−0.384i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1053s/2 ΓR(s) L(s)=((0.923−0.384i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10531053    =    34⋅133^{4} \cdot 13
Sign: 0.923−0.384i0.923 - 0.384i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(119,⋅)\chi_{1053} (119, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), 0.923−0.384i)(1,\ 1053,\ (0:\ ),\ 0.923 - 0.384i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.348432163−0.4690237977i2.348432163 - 0.4690237977i
L(12)L(\frac12) ≈\approx 2.348432163−0.4690237977i2.348432163 - 0.4690237977i
L(1)L(1) ≈\approx 1.666328817−0.1804865285i1.666328817 - 0.1804865285i
L(1)L(1) ≈\approx 1.666328817−0.1804865285i1.666328817 - 0.1804865285i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1 1
good2 1+(0.998−0.0581i)T 1 + (0.998 - 0.0581i)T
5 1+(−0.727−0.686i)T 1 + (-0.727 - 0.686i)T
7 1+(−0.918+0.396i)T 1 + (-0.918 + 0.396i)T
11 1+(−0.727+0.686i)T 1 + (-0.727 + 0.686i)T
17 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
19 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
23 1+(0.396−0.918i)T 1 + (0.396 - 0.918i)T
29 1+(0.835−0.549i)T 1 + (0.835 - 0.549i)T
31 1+(0.802−0.597i)T 1 + (0.802 - 0.597i)T
37 1+(−0.342−0.939i)T 1 + (-0.342 - 0.939i)T
41 1+(0.448+0.893i)T 1 + (0.448 + 0.893i)T
43 1+(0.686+0.727i)T 1 + (0.686 + 0.727i)T
47 1+(−0.802−0.597i)T 1 + (-0.802 - 0.597i)T
53 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
59 1+(0.727+0.686i)T 1 + (0.727 + 0.686i)T
61 1+(0.597−0.802i)T 1 + (0.597 - 0.802i)T
67 1+(0.998+0.0581i)T 1 + (0.998 + 0.0581i)T
71 1+(−0.342−0.939i)T 1 + (-0.342 - 0.939i)T
73 1+(−0.984+0.173i)T 1 + (-0.984 + 0.173i)T
79 1+(−0.835+0.549i)T 1 + (-0.835 + 0.549i)T
83 1+(−0.549−0.835i)T 1 + (-0.549 - 0.835i)T
89 1+(−0.342+0.939i)T 1 + (-0.342 + 0.939i)T
97 1+(0.957+0.286i)T 1 + (0.957 + 0.286i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−21.76032146847913151571046103503, −20.86041670700214421787206005133, −20.12189264084660357452677997595, −19.237203430101000289466047888, −18.8406305525783503783210634276, −17.61216767411804561709482511726, −16.44612096181463851347542663130, −15.87464486422171723446127755269, −15.49213524591283554077340273403, −14.20707829513695552378547091541, −13.87904728881306878300141856375, −12.93653983375978114495971126840, −12.06822438335890202943289131283, −11.42749168148847796597737209242, −10.51161964594367709772091560957, −9.9036266365745140352608695063, −8.409383142217811678762662251234, −7.39308679912977652030372952464, −7.00682903699451204092569052164, −5.937818665660874172217231384401, −5.15085676110901891032494597151, −4.00037363190758779722581275112, −3.07715595440915531389538382347, −2.89561999429618714264116007031, −1.020305898044660688520119461807, 0.91813665147221518203904033253, 2.359296874221942313165723531667, 3.18162786415190825120138132717, 4.1061297244070840922559564057, 4.94342616716978210728663724869, 5.71837844367373801230089316360, 6.68724950766377853592948411630, 7.59631755149672516418747806313, 8.355220752117726184798688529121, 9.63404727486169683277552233972, 10.31441326215159818947404829616, 11.47629213664592357313324483567, 12.16035157535824702713406822231, 12.80502203601513303238669202431, 13.27319888161507708661983357030, 14.48871378711015966765801981264, 15.2156032045952004483609910755, 16.000341704335105155669582020654, 16.30813321488212743751232352727, 17.38813524399753669231959842748, 18.631208814341101051128844114175, 19.43369159301398162963960622679, 19.95982096086931837843340095181, 20.88387622023031863679644351742, 21.32581421357585058275367812630

Graph of the ZZ-function along the critical line