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Properties

Label 2-161-161.103-c1-0-5
Degree 22
Conductor 161161
Sign −0.242−0.970i-0.242 - 0.970i
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  + (0.0675 + 1.41i)2-s + (0.681 + 1.70i)3-s + (−0.0160 + 0.00153i)4-s + (0.132 − 0.546i)5-s + (−2.36 + 1.08i)6-s + (2.36 − 1.17i)7-s + (0.400 + 2.78i)8-s + (−0.258 + 0.246i)9-s + (0.783 + 0.151i)10-s + (−5.46 − 0.260i)11-s + (−0.0135 − 0.0262i)12-s + (−3.18 − 2.76i)13-s + (1.83 + 3.27i)14-s + (1.01 − 0.146i)15-s + (−3.95 + 0.763i)16-s + (2.00 − 2.81i)17-s + ⋯
L(s)  = 1  + (0.0477 + 1.00i)2-s + (0.393 + 0.982i)3-s + (−0.00801 + 0.000765i)4-s + (0.0592 − 0.244i)5-s + (−0.966 + 0.441i)6-s + (0.895 − 0.445i)7-s + (0.141 + 0.985i)8-s + (−0.0863 + 0.0823i)9-s + (0.247 + 0.0477i)10-s + (−1.64 − 0.0785i)11-s + (−0.00390 − 0.00757i)12-s + (−0.884 − 0.766i)13-s + (0.489 + 0.876i)14-s + (0.263 − 0.0378i)15-s + (−0.989 + 0.190i)16-s + (0.485 − 0.682i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.242−0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((−0.242−0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.242−0.970i-0.242 - 0.970i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(103,⋅)\chi_{161} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), −0.242−0.970i)(2,\ 161,\ (\ :1/2),\ -0.242 - 0.970i)

Particular Values

L(1)L(1) ≈\approx 0.884964+1.13308i0.884964 + 1.13308i
L(12)L(\frac12) ≈\approx 0.884964+1.13308i0.884964 + 1.13308i
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+(−2.36+1.17i)T 1 + (-2.36 + 1.17i)T
23 1+(4.58+1.41i)T 1 + (4.58 + 1.41i)T
good2 1+(−0.0675−1.41i)T+(−1.99+0.190i)T2 1 + (-0.0675 - 1.41i)T + (-1.99 + 0.190i)T^{2}
3 1+(−0.681−1.70i)T+(−2.17+2.07i)T2 1 + (-0.681 - 1.70i)T + (-2.17 + 2.07i)T^{2}
5 1+(−0.132+0.546i)T+(−4.44−2.29i)T2 1 + (-0.132 + 0.546i)T + (-4.44 - 2.29i)T^{2}
11 1+(5.46+0.260i)T+(10.9+1.04i)T2 1 + (5.46 + 0.260i)T + (10.9 + 1.04i)T^{2}
13 1+(3.18+2.76i)T+(1.85+12.8i)T2 1 + (3.18 + 2.76i)T + (1.85 + 12.8i)T^{2}
17 1+(−2.00+2.81i)T+(−5.56−16.0i)T2 1 + (-2.00 + 2.81i)T + (-5.56 - 16.0i)T^{2}
19 1+(2.51+3.53i)T+(−6.21+17.9i)T2 1 + (2.51 + 3.53i)T + (-6.21 + 17.9i)T^{2}
29 1+(−3.66−8.02i)T+(−18.9+21.9i)T2 1 + (-3.66 - 8.02i)T + (-18.9 + 21.9i)T^{2}
31 1+(−2.37−3.01i)T+(−7.30+30.1i)T2 1 + (-2.37 - 3.01i)T + (-7.30 + 30.1i)T^{2}
37 1+(1.88+1.97i)T+(−1.76+36.9i)T2 1 + (1.88 + 1.97i)T + (-1.76 + 36.9i)T^{2}
41 1+(0.322+1.09i)T+(−34.4+22.1i)T2 1 + (0.322 + 1.09i)T + (-34.4 + 22.1i)T^{2}
43 1+(−3.95−0.569i)T+(41.2+12.1i)T2 1 + (-3.95 - 0.569i)T + (41.2 + 12.1i)T^{2}
47 1+(9.85+5.69i)T+(23.5+40.7i)T2 1 + (9.85 + 5.69i)T + (23.5 + 40.7i)T^{2}
53 1+(−4.73−1.63i)T+(41.6+32.7i)T2 1 + (-4.73 - 1.63i)T + (41.6 + 32.7i)T^{2}
59 1+(0.461−2.39i)T+(−54.7−21.9i)T2 1 + (0.461 - 2.39i)T + (-54.7 - 21.9i)T^{2}
61 1+(3.12+1.24i)T+(44.1+42.0i)T2 1 + (3.12 + 1.24i)T + (44.1 + 42.0i)T^{2}
67 1+(5.57−10.8i)T+(−38.8−54.5i)T2 1 + (5.57 - 10.8i)T + (-38.8 - 54.5i)T^{2}
71 1+(6.13−3.94i)T+(29.4−64.5i)T2 1 + (6.13 - 3.94i)T + (29.4 - 64.5i)T^{2}
73 1+(1.25+13.1i)T+(−71.6+13.8i)T2 1 + (1.25 + 13.1i)T + (-71.6 + 13.8i)T^{2}
79 1+(−2.77+0.959i)T+(62.0−48.8i)T2 1 + (-2.77 + 0.959i)T + (62.0 - 48.8i)T^{2}
83 1+(−11.4−3.35i)T+(69.8+44.8i)T2 1 + (-11.4 - 3.35i)T + (69.8 + 44.8i)T^{2}
89 1+(−9.02−7.09i)T+(20.9+86.4i)T2 1 + (-9.02 - 7.09i)T + (20.9 + 86.4i)T^{2}
97 1+(1.24−0.366i)T+(81.6−52.4i)T2 1 + (1.24 - 0.366i)T + (81.6 - 52.4i)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

−13.50140678152572218808977727334, −12.23531042059484455227786379877, −10.73589917224579451385847886764, −10.30905835791723701622715868606, −8.800914279748822747392873575792, −7.931472185085633391804651623982, −7.04513696475538352159784698452, −5.20539245090189676669917522761, −4.83346112697645214729260295520, −2.78957808754961203802807920378, 1.88391959859922651964723193619, 2.61265757931663118713140825385, 4.56237025493283408911495915496, 6.25731454680139484145647250272, 7.63066059838072223930144377719, 8.166363441420690044751665845414, 9.943604776348644704512322078893, 10.63262286199046345626799866890, 11.85475339008191312920890942153, 12.41634933788821775438486641752

Graph of the ZZ-function along the critical line