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Properties

Label 1-161-161.136-r0-0-0
Degree 11
Conductor 161161
Sign −0.0815−0.996i-0.0815 - 0.996i
Analytic cond. 0.7476800.747680
Root an. cond. 0.7476800.747680
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  + (−0.995 − 0.0950i)2-s + (−0.723 − 0.690i)3-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)10-s + (0.995 − 0.0950i)11-s + (−0.580 − 0.814i)12-s + (0.142 − 0.989i)13-s + (0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.0475 − 0.998i)18-s + (−0.327 − 0.945i)19-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (−0.723 − 0.690i)3-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)10-s + (0.995 − 0.0950i)11-s + (−0.580 − 0.814i)12-s + (0.142 − 0.989i)13-s + (0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.0475 − 0.998i)18-s + (−0.327 − 0.945i)19-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓR(s) L(s)=((−0.0815−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0815 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(161s/2 ΓR(s) L(s)=((−0.0815−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0815 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.0815−0.996i-0.0815 - 0.996i
Analytic conductor: 0.7476800.747680
Root analytic conductor: 0.7476800.747680
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ161(136,⋅)\chi_{161} (136, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 161, (0: ), −0.0815−0.996i)(1,\ 161,\ (0:\ ),\ -0.0815 - 0.996i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.2745789834−0.2979490508i0.2745789834 - 0.2979490508i
L(12)L(\frac12) ≈\approx 0.2745789834−0.2979490508i0.2745789834 - 0.2979490508i
L(1)L(1) ≈\approx 0.4576732473−0.1538113325i0.4576732473 - 0.1538113325i
L(1)L(1) ≈\approx 0.4576732473−0.1538113325i0.4576732473 - 0.1538113325i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
23 1 1
good2 1+(−0.995−0.0950i)T 1 + (-0.995 - 0.0950i)T
3 1+(−0.723−0.690i)T 1 + (-0.723 - 0.690i)T
5 1+(−0.888+0.458i)T 1 + (-0.888 + 0.458i)T
11 1+(0.995−0.0950i)T 1 + (0.995 - 0.0950i)T
13 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
17 1+(−0.327+0.945i)T 1 + (-0.327 + 0.945i)T
19 1+(−0.327−0.945i)T 1 + (-0.327 - 0.945i)T
29 1+(−0.654−0.755i)T 1 + (-0.654 - 0.755i)T
31 1+(−0.235−0.971i)T 1 + (-0.235 - 0.971i)T
37 1+(−0.0475−0.998i)T 1 + (-0.0475 - 0.998i)T
41 1+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
43 1+(0.959−0.281i)T 1 + (0.959 - 0.281i)T
47 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
53 1+(0.786−0.618i)T 1 + (0.786 - 0.618i)T
59 1+(−0.928+0.371i)T 1 + (-0.928 + 0.371i)T
61 1+(0.723−0.690i)T 1 + (0.723 - 0.690i)T
67 1+(−0.580+0.814i)T 1 + (-0.580 + 0.814i)T
71 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
73 1+(−0.981−0.189i)T 1 + (-0.981 - 0.189i)T
79 1+(0.786+0.618i)T 1 + (0.786 + 0.618i)T
83 1+(0.841−0.540i)T 1 + (0.841 - 0.540i)T
89 1+(0.235−0.971i)T 1 + (0.235 - 0.971i)T
97 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)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

−27.74036434296809182382614594429, −27.31027323535749743713481143988, −26.510928831918552504449354551101, −25.25010243918114754501347311086, −24.16083151894981002190208681085, −23.34683356533391496730468713135, −22.191450919662980655268933339751, −20.94257562017645089675524405120, −20.18045110167918685414797864570, −19.12433733595367986098330887133, −18.10867623238777158000600578319, −16.81103308944456214248133128628, −16.46444240305240108670531414099, −15.482067934236538200663945266362, −14.452505869202351627903703824907, −12.24671865631844776178419124038, −11.66626483158079225778910493501, −10.751827993359150168927560318750, −9.42037058013615129916636313000, −8.77332321543281809003490627474, −7.272156451621687440186051737088, −6.273168037022245064345361881, −4.755588230732013759871360176714, −3.537385353684328299179232018078, −1.3059212892926127713960163271, 0.574130428886037437041555104960, 2.22421344364273220874776297593, 3.84916409392407975166180265114, 5.876444277551218470162165659662, 6.881986890401253904767311830, 7.747360178237771600154151434050, 8.790461475861589150933148605901, 10.43269245444688163012399868191, 11.19890151953836363599045067867, 11.99489256147380427706327247280, 13.06152906309910498938665463211, 14.866470801228553247968362679346, 15.75822022662905136070583386375, 16.97797981828760725060457579859, 17.61568791292068856755235982967, 18.71397958824536079592023446330, 19.42438349781860246362202282061, 20.1663850444177844576333128171, 21.83618638678019752512889966901, 22.704294299046509847507936389795, 23.87505569596736260040910942258, 24.59547104004471871981254053474, 25.68331170343755101187270388644, 26.73585902688445034197569899413, 27.864120602426276807093420431522

Graph of the ZZ-function along the critical line