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Properties

Label 2-161-161.10-c1-0-1
Degree 22
Conductor 161161
Sign −0.988−0.148i-0.988 - 0.148i
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  + (−0.601 + 1.73i)2-s + (0.530 + 1.02i)3-s + (−1.09 − 0.857i)4-s + (−1.45 + 0.138i)5-s + (−2.10 + 0.303i)6-s + (−2.03 + 1.69i)7-s + (−0.948 + 0.609i)8-s + (0.962 − 1.35i)9-s + (0.633 − 2.61i)10-s + (−0.0712 + 0.0246i)11-s + (0.303 − 1.57i)12-s + (1.89 + 6.45i)13-s + (−1.72 − 4.55i)14-s + (−0.913 − 1.42i)15-s + (−1.14 − 4.71i)16-s + (−0.122 − 0.0489i)17-s + ⋯
L(s)  = 1  + (−0.425 + 1.22i)2-s + (0.306 + 0.594i)3-s + (−0.545 − 0.428i)4-s + (−0.649 + 0.0620i)5-s + (−0.860 + 0.123i)6-s + (−0.767 + 0.640i)7-s + (−0.335 + 0.215i)8-s + (0.320 − 0.450i)9-s + (0.200 − 0.825i)10-s + (−0.0214 + 0.00743i)11-s + (0.0877 − 0.455i)12-s + (0.525 + 1.79i)13-s + (−0.461 − 1.21i)14-s + (−0.235 − 0.366i)15-s + (−0.285 − 1.17i)16-s + (−0.0296 − 0.0118i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.988−0.148i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((−0.988−0.148i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.988−0.148i-0.988 - 0.148i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(10,⋅)\chi_{161} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), −0.988−0.148i)(2,\ 161,\ (\ :1/2),\ -0.988 - 0.148i)

Particular Values

L(1)L(1) ≈\approx 0.0614640+0.825012i0.0614640 + 0.825012i
L(12)L(\frac12) ≈\approx 0.0614640+0.825012i0.0614640 + 0.825012i
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+(2.03−1.69i)T 1 + (2.03 - 1.69i)T
23 1+(3.15+3.61i)T 1 + (3.15 + 3.61i)T
good2 1+(0.601−1.73i)T+(−1.57−1.23i)T2 1 + (0.601 - 1.73i)T + (-1.57 - 1.23i)T^{2}
3 1+(−0.530−1.02i)T+(−1.74+2.44i)T2 1 + (-0.530 - 1.02i)T + (-1.74 + 2.44i)T^{2}
5 1+(1.45−0.138i)T+(4.90−0.946i)T2 1 + (1.45 - 0.138i)T + (4.90 - 0.946i)T^{2}
11 1+(0.0712−0.0246i)T+(8.64−6.79i)T2 1 + (0.0712 - 0.0246i)T + (8.64 - 6.79i)T^{2}
13 1+(−1.89−6.45i)T+(−10.9+7.02i)T2 1 + (-1.89 - 6.45i)T + (-10.9 + 7.02i)T^{2}
17 1+(0.122+0.0489i)T+(12.3+11.7i)T2 1 + (0.122 + 0.0489i)T + (12.3 + 11.7i)T^{2}
19 1+(−6.22+2.49i)T+(13.7−13.1i)T2 1 + (-6.22 + 2.49i)T + (13.7 - 13.1i)T^{2}
29 1+(−1.08−7.51i)T+(−27.8+8.17i)T2 1 + (-1.08 - 7.51i)T + (-27.8 + 8.17i)T^{2}
31 1+(−8.46+0.403i)T+(30.8−2.94i)T2 1 + (-8.46 + 0.403i)T + (30.8 - 2.94i)T^{2}
37 1+(−4.96−3.53i)T+(12.1+34.9i)T2 1 + (-4.96 - 3.53i)T + (12.1 + 34.9i)T^{2}
41 1+(−0.923−0.421i)T+(26.8+30.9i)T2 1 + (-0.923 - 0.421i)T + (26.8 + 30.9i)T^{2}
43 1+(2.48−3.86i)T+(−17.8−39.1i)T2 1 + (2.48 - 3.86i)T + (-17.8 - 39.1i)T^{2}
47 1+(0.735−0.424i)T+(23.5−40.7i)T2 1 + (0.735 - 0.424i)T + (23.5 - 40.7i)T^{2}
53 1+(6.26+6.57i)T+(−2.52+52.9i)T2 1 + (6.26 + 6.57i)T + (-2.52 + 52.9i)T^{2}
59 1+(−5.69−1.38i)T+(52.4+27.0i)T2 1 + (-5.69 - 1.38i)T + (52.4 + 27.0i)T^{2}
61 1+(1.85+0.957i)T+(35.3+49.6i)T2 1 + (1.85 + 0.957i)T + (35.3 + 49.6i)T^{2}
67 1+(2.00+10.3i)T+(−62.2+24.9i)T2 1 + (2.00 + 10.3i)T + (-62.2 + 24.9i)T^{2}
71 1+(2.67+3.08i)T+(−10.1+70.2i)T2 1 + (2.67 + 3.08i)T + (-10.1 + 70.2i)T^{2}
73 1+(−0.126+0.160i)T+(−17.2−70.9i)T2 1 + (-0.126 + 0.160i)T + (-17.2 - 70.9i)T^{2}
79 1+(−4.20+4.40i)T+(−3.75−78.9i)T2 1 + (-4.20 + 4.40i)T + (-3.75 - 78.9i)T^{2}
83 1+(4.32+9.47i)T+(−54.3+62.7i)T2 1 + (4.32 + 9.47i)T + (-54.3 + 62.7i)T^{2}
89 1+(0.572−12.0i)T+(−88.5−8.45i)T2 1 + (0.572 - 12.0i)T + (-88.5 - 8.45i)T^{2}
97 1+(0.934−2.04i)T+(−63.5−73.3i)T2 1 + (0.934 - 2.04i)T + (-63.5 - 73.3i)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

−13.73808802523294591372169199502, −12.17572491446894516427831760901, −11.51852609650035285899008910598, −9.743808544752799041819930947853, −9.168437006637021286143972857834, −8.253184921928359619208424539453, −6.93662293134136347444805053019, −6.27820590993704384633329865643, −4.63389646948726101146096958465, −3.20636708761911581915962979194, 0.935131479174652699274597432744, 2.82915703925132000836328278093, 3.88817646950641330017945664041, 5.98065274108846857437833468019, 7.54416040598786220170437131269, 8.180416400824753117572401901969, 9.828372405496682667939735731783, 10.27158276183574085101598203279, 11.44748291906350347793702855161, 12.30871624546155329949727955291

Graph of the ZZ-function along the critical line