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Properties

Label 2-161-7.4-c3-0-11
Degree 22
Conductor 161161
Sign −0.864−0.502i-0.864 - 0.502i
Analytic cond. 9.499309.49930
Root an. cond. 3.082093.08209
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.81 + 3.13i)2-s + (−0.112 + 0.195i)3-s + (−2.56 + 4.44i)4-s + (−1.55 − 2.69i)5-s − 0.817·6-s + (−8.28 + 16.5i)7-s + 10.3·8-s + (13.4 + 23.3i)9-s + (5.64 − 9.77i)10-s + (−13.6 + 23.6i)11-s + (−0.579 − 1.00i)12-s − 46.9·13-s + (−66.9 + 4.02i)14-s + 0.702·15-s + (39.3 + 68.1i)16-s + (−30.3 + 52.6i)17-s + ⋯
L(s)  = 1  + (0.640 + 1.10i)2-s + (−0.0217 + 0.0376i)3-s + (−0.320 + 0.555i)4-s + (−0.139 − 0.241i)5-s − 0.0556·6-s + (−0.447 + 0.894i)7-s + 0.459·8-s + (0.499 + 0.864i)9-s + (0.178 − 0.308i)10-s + (−0.374 + 0.648i)11-s + (−0.0139 − 0.0241i)12-s − 1.00·13-s + (−1.27 + 0.0769i)14-s + 0.0120·15-s + (0.614 + 1.06i)16-s + (−0.433 + 0.750i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.864−0.502i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((−0.864−0.502i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.864−0.502i-0.864 - 0.502i
Analytic conductor: 9.499309.49930
Root analytic conductor: 3.082093.08209
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ161(116,⋅)\chi_{161} (116, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :3/2), −0.864−0.502i)(2,\ 161,\ (\ :3/2),\ -0.864 - 0.502i)

Particular Values

L(2)L(2) ≈\approx 0.529194+1.96199i0.529194 + 1.96199i
L(12)L(\frac12) ≈\approx 0.529194+1.96199i0.529194 + 1.96199i
L(52)L(\frac{5}{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+(8.28−16.5i)T 1 + (8.28 - 16.5i)T
23 1+(−11.5−19.9i)T 1 + (-11.5 - 19.9i)T
good2 1+(−1.81−3.13i)T+(−4+6.92i)T2 1 + (-1.81 - 3.13i)T + (-4 + 6.92i)T^{2}
3 1+(0.112−0.195i)T+(−13.5−23.3i)T2 1 + (0.112 - 0.195i)T + (-13.5 - 23.3i)T^{2}
5 1+(1.55+2.69i)T+(−62.5+108.i)T2 1 + (1.55 + 2.69i)T + (-62.5 + 108. i)T^{2}
11 1+(13.6−23.6i)T+(−665.5−1.15e3i)T2 1 + (13.6 - 23.6i)T + (-665.5 - 1.15e3i)T^{2}
13 1+46.9T+2.19e3T2 1 + 46.9T + 2.19e3T^{2}
17 1+(30.3−52.6i)T+(−2.45e3−4.25e3i)T2 1 + (30.3 - 52.6i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(0.916+1.58i)T+(−3.42e3+5.94e3i)T2 1 + (0.916 + 1.58i)T + (-3.42e3 + 5.94e3i)T^{2}
29 1−115.T+2.43e4T2 1 - 115.T + 2.43e4T^{2}
31 1+(−32.9+57.0i)T+(−1.48e4−2.57e4i)T2 1 + (-32.9 + 57.0i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(4.26+7.38i)T+(−2.53e4+4.38e4i)T2 1 + (4.26 + 7.38i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−129.T+6.89e4T2 1 - 129.T + 6.89e4T^{2}
43 1−200.T+7.95e4T2 1 - 200.T + 7.95e4T^{2}
47 1+(−38.8−67.3i)T+(−5.19e4+8.99e4i)T2 1 + (-38.8 - 67.3i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−14.2+24.6i)T+(−7.44e4−1.28e5i)T2 1 + (-14.2 + 24.6i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(25.6−44.3i)T+(−1.02e5−1.77e5i)T2 1 + (25.6 - 44.3i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−30.5−52.9i)T+(−1.13e5+1.96e5i)T2 1 + (-30.5 - 52.9i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−8.64+14.9i)T+(−1.50e5−2.60e5i)T2 1 + (-8.64 + 14.9i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−366.T+3.57e5T2 1 - 366.T + 3.57e5T^{2}
73 1+(−455.+788.i)T+(−1.94e5−3.36e5i)T2 1 + (-455. + 788. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(−611.−1.05e3i)T+(−2.46e5+4.26e5i)T2 1 + (-611. - 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−825.T+5.71e5T2 1 - 825.T + 5.71e5T^{2}
89 1+(348.+604.i)T+(−3.52e5+6.10e5i)T2 1 + (348. + 604. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+658.T+9.12e5T2 1 + 658.T + 9.12e5T^{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

−12.87416676149259432609094433513, −12.33115433582182043296008821400, −10.73800243837981416363099749909, −9.753748314799284560336951920450, −8.349172650189480176984421210863, −7.42467980597244703581058791379, −6.37350196119753679946133759786, −5.20517564707635295621585456862, −4.41931712490504885616502861771, −2.27029702687444827053645783945, 0.792845145145772000889433210846, 2.74849234956810982933598187025, 3.76918875129613470907039213603, 4.91120153520309800844354696173, 6.70424173718109941940030815110, 7.59275909589948790080300945082, 9.348687653089069348374132590516, 10.28680381978303728249281129481, 11.07909636723422349360340240060, 12.08541874580112826017010394788

Graph of the ZZ-function along the critical line