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Properties

Label 2-161-161.100-c1-0-5
Degree 22
Conductor 161161
Sign −0.102−0.994i-0.102 - 0.994i
Analytic cond. 1.285591.28559
Root an. cond. 1.133831.13383
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  + (1.92 + 1.51i)2-s + (−0.261 + 0.367i)3-s + (0.941 + 3.88i)4-s + (−1.46 + 0.281i)5-s + (−1.06 + 0.311i)6-s + (−2.01 − 1.70i)7-s + (−2.02 + 4.43i)8-s + (0.914 + 2.64i)9-s + (−3.24 − 1.67i)10-s + (4.03 − 3.17i)11-s + (−1.67 − 0.670i)12-s + (5.59 − 3.59i)13-s + (−1.30 − 6.34i)14-s + (0.279 − 0.611i)15-s + (−3.52 + 1.81i)16-s + (−1.75 − 1.66i)17-s + ⋯
L(s)  = 1  + (1.36 + 1.07i)2-s + (−0.151 + 0.212i)3-s + (0.470 + 1.94i)4-s + (−0.654 + 0.126i)5-s + (−0.432 + 0.127i)6-s + (−0.763 − 0.645i)7-s + (−0.716 + 1.56i)8-s + (0.304 + 0.880i)9-s + (−1.02 − 0.528i)10-s + (1.21 − 0.956i)11-s + (−0.483 − 0.193i)12-s + (1.55 − 0.997i)13-s + (−0.347 − 1.69i)14-s + (0.0721 − 0.157i)15-s + (−0.880 + 0.453i)16-s + (−0.424 − 0.404i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.102−0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((−0.102−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.102−0.994i-0.102 - 0.994i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(100,⋅)\chi_{161} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), −0.102−0.994i)(2,\ 161,\ (\ :1/2),\ -0.102 - 0.994i)

Particular Values

L(1)L(1) ≈\approx 1.28752+1.42722i1.28752 + 1.42722i
L(12)L(\frac12) ≈\approx 1.28752+1.42722i1.28752 + 1.42722i
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)
bad7 1+(2.01+1.70i)T 1 + (2.01 + 1.70i)T
23 1+(4.13+2.43i)T 1 + (4.13 + 2.43i)T
good2 1+(−1.92−1.51i)T+(0.471+1.94i)T2 1 + (-1.92 - 1.51i)T + (0.471 + 1.94i)T^{2}
3 1+(0.261−0.367i)T+(−0.981−2.83i)T2 1 + (0.261 - 0.367i)T + (-0.981 - 2.83i)T^{2}
5 1+(1.46−0.281i)T+(4.64−1.85i)T2 1 + (1.46 - 0.281i)T + (4.64 - 1.85i)T^{2}
11 1+(−4.03+3.17i)T+(2.59−10.6i)T2 1 + (-4.03 + 3.17i)T + (2.59 - 10.6i)T^{2}
13 1+(−5.59+3.59i)T+(5.40−11.8i)T2 1 + (-5.59 + 3.59i)T + (5.40 - 11.8i)T^{2}
17 1+(1.75+1.66i)T+(0.808+16.9i)T2 1 + (1.75 + 1.66i)T + (0.808 + 16.9i)T^{2}
19 1+(4.76−4.54i)T+(0.904−18.9i)T2 1 + (4.76 - 4.54i)T + (0.904 - 18.9i)T^{2}
29 1+(3.87−1.13i)T+(24.3−15.6i)T2 1 + (3.87 - 1.13i)T + (24.3 - 15.6i)T^{2}
31 1+(1.22−0.116i)T+(30.4−5.86i)T2 1 + (1.22 - 0.116i)T + (30.4 - 5.86i)T^{2}
37 1+(0.215+0.622i)T+(−29.0+22.8i)T2 1 + (0.215 + 0.622i)T + (-29.0 + 22.8i)T^{2}
41 1+(0.187+0.216i)T+(−5.83+40.5i)T2 1 + (0.187 + 0.216i)T + (-5.83 + 40.5i)T^{2}
43 1+(0.936+2.04i)T+(−28.1+32.4i)T2 1 + (0.936 + 2.04i)T + (-28.1 + 32.4i)T^{2}
47 1+(−3.09+5.35i)T+(−23.5−40.7i)T2 1 + (-3.09 + 5.35i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.199−4.18i)T+(−52.7−5.03i)T2 1 + (0.199 - 4.18i)T + (-52.7 - 5.03i)T^{2}
59 1+(0.755+0.389i)T+(34.2+48.0i)T2 1 + (0.755 + 0.389i)T + (34.2 + 48.0i)T^{2}
61 1+(−5.51−7.74i)T+(−19.9+57.6i)T2 1 + (-5.51 - 7.74i)T + (-19.9 + 57.6i)T^{2}
67 1+(−2.12+0.852i)T+(48.4−46.2i)T2 1 + (-2.12 + 0.852i)T + (48.4 - 46.2i)T^{2}
71 1+(−1.78+12.3i)T+(−68.1−20.0i)T2 1 + (-1.78 + 12.3i)T + (-68.1 - 20.0i)T^{2}
73 1+(−0.738−3.04i)T+(−64.8+33.4i)T2 1 + (-0.738 - 3.04i)T + (-64.8 + 33.4i)T^{2}
79 1+(0.00251+0.0528i)T+(−78.6+7.50i)T2 1 + (0.00251 + 0.0528i)T + (-78.6 + 7.50i)T^{2}
83 1+(4.18−4.82i)T+(−11.8−82.1i)T2 1 + (4.18 - 4.82i)T + (-11.8 - 82.1i)T^{2}
89 1+(5.34+0.510i)T+(87.3+16.8i)T2 1 + (5.34 + 0.510i)T + (87.3 + 16.8i)T^{2}
97 1+(−9.58−11.0i)T+(−13.8+96.0i)T2 1 + (-9.58 - 11.0i)T + (-13.8 + 96.0i)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

−13.44183973183833374569314884003, −12.54595552509602636052692813132, −11.35270031566691023831846959655, −10.42577295340623079782697730883, −8.558764345204106602572228070240, −7.64169060800788805479060554527, −6.46355786833951885982459230387, −5.74517383353352792415677352528, −4.03204978482297457595226411379, −3.65086791590542915551626357582, 1.84293384230217039764859118603, 3.76378036614209192194091357211, 4.20672916733194856547963347659, 6.10120930579645407258255972457, 6.67803170338506635062232091846, 8.858428822026022097297610855904, 9.754805528894548042556769534892, 11.29048129619331070588223219647, 11.72893180805479263932604830396, 12.59456610394135128107431358334

Graph of the ZZ-function along the critical line