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Properties

Label 2-171-171.110-c1-0-17
Degree 22
Conductor 171171
Sign −0.424+0.905i-0.424 + 0.905i
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
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  + (1.98 − 1.66i)2-s + (−0.713 − 1.57i)3-s + (0.819 − 4.64i)4-s + (−0.441 + 1.21i)5-s + (−4.04 − 1.94i)6-s + (2.16 + 3.74i)7-s + (−3.52 − 6.10i)8-s + (−1.98 + 2.25i)9-s + (1.14 + 3.14i)10-s + 2.72i·11-s + (−7.92 + 2.02i)12-s + (−1.29 − 3.55i)13-s + (10.5 + 3.83i)14-s + (2.23 − 0.169i)15-s + (−8.30 − 3.02i)16-s + (0.00982 − 0.0269i)17-s + ⋯
L(s)  = 1  + (1.40 − 1.17i)2-s + (−0.412 − 0.911i)3-s + (0.409 − 2.32i)4-s + (−0.197 + 0.542i)5-s + (−1.65 − 0.793i)6-s + (0.817 + 1.41i)7-s + (−1.24 − 2.15i)8-s + (−0.660 + 0.751i)9-s + (0.362 + 0.995i)10-s + 0.822i·11-s + (−2.28 + 0.584i)12-s + (−0.359 − 0.987i)13-s + (2.81 + 1.02i)14-s + (0.576 − 0.0437i)15-s + (−2.07 − 0.756i)16-s + (0.00238 − 0.00654i)17-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((−0.424+0.905i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((−0.424+0.905i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: −0.424+0.905i-0.424 + 0.905i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ171(110,⋅)\chi_{171} (110, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), −0.424+0.905i)(2,\ 171,\ (\ :1/2),\ -0.424 + 0.905i)

Particular Values

L(1)L(1) ≈\approx 1.07508−1.69099i1.07508 - 1.69099i
L(12)L(\frac12) ≈\approx 1.07508−1.69099i1.07508 - 1.69099i
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)
bad3 1+(0.713+1.57i)T 1 + (0.713 + 1.57i)T
19 1+(0.236+4.35i)T 1 + (0.236 + 4.35i)T
good2 1+(−1.98+1.66i)T+(0.347−1.96i)T2 1 + (-1.98 + 1.66i)T + (0.347 - 1.96i)T^{2}
5 1+(0.441−1.21i)T+(−3.83−3.21i)T2 1 + (0.441 - 1.21i)T + (-3.83 - 3.21i)T^{2}
7 1+(−2.16−3.74i)T+(−3.5+6.06i)T2 1 + (-2.16 - 3.74i)T + (-3.5 + 6.06i)T^{2}
11 1−2.72iT−11T2 1 - 2.72iT - 11T^{2}
13 1+(1.29+3.55i)T+(−9.95+8.35i)T2 1 + (1.29 + 3.55i)T + (-9.95 + 8.35i)T^{2}
17 1+(−0.00982+0.0269i)T+(−13.0−10.9i)T2 1 + (-0.00982 + 0.0269i)T + (-13.0 - 10.9i)T^{2}
23 1+(5.97+1.05i)T+(21.6+7.86i)T2 1 + (5.97 + 1.05i)T + (21.6 + 7.86i)T^{2}
29 1+(0.592−3.35i)T+(−27.2−9.91i)T2 1 + (0.592 - 3.35i)T + (-27.2 - 9.91i)T^{2}
31 1+3.60iT−31T2 1 + 3.60iT - 31T^{2}
37 1−7.25iT−37T2 1 - 7.25iT - 37T^{2}
41 1+(2.90−2.43i)T+(7.11−40.3i)T2 1 + (2.90 - 2.43i)T + (7.11 - 40.3i)T^{2}
43 1+(0.543+3.08i)T+(−40.4+14.7i)T2 1 + (0.543 + 3.08i)T + (-40.4 + 14.7i)T^{2}
47 1+(7.07+1.24i)T+(44.1+16.0i)T2 1 + (7.07 + 1.24i)T + (44.1 + 16.0i)T^{2}
53 1+(−4.24−3.56i)T+(9.20+52.1i)T2 1 + (-4.24 - 3.56i)T + (9.20 + 52.1i)T^{2}
59 1+(1.88+10.6i)T+(−55.4+20.1i)T2 1 + (1.88 + 10.6i)T + (-55.4 + 20.1i)T^{2}
61 1+(−9.95+3.62i)T+(46.7−39.2i)T2 1 + (-9.95 + 3.62i)T + (46.7 - 39.2i)T^{2}
67 1+(−6.64+7.91i)T+(−11.6−65.9i)T2 1 + (-6.64 + 7.91i)T + (-11.6 - 65.9i)T^{2}
71 1+(4.93−4.13i)T+(12.3−69.9i)T2 1 + (4.93 - 4.13i)T + (12.3 - 69.9i)T^{2}
73 1+(−1.86−10.5i)T+(−68.5+24.9i)T2 1 + (-1.86 - 10.5i)T + (-68.5 + 24.9i)T^{2}
79 1+(1.76−4.86i)T+(−60.5−50.7i)T2 1 + (1.76 - 4.86i)T + (-60.5 - 50.7i)T^{2}
83 1+(7.70−4.44i)T+(41.5−71.8i)T2 1 + (7.70 - 4.44i)T + (41.5 - 71.8i)T^{2}
89 1+(−1.81+10.2i)T+(−83.6−30.4i)T2 1 + (-1.81 + 10.2i)T + (-83.6 - 30.4i)T^{2}
97 1+(0.845+1.00i)T+(−16.8+95.5i)T2 1 + (0.845 + 1.00i)T + (-16.8 + 95.5i)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

−12.44115334775737824001275167297, −11.65499890247807057306865878406, −11.10936798843308199860817145375, −9.938712594682753627633043183589, −8.238584297944481069611378899553, −6.77174823297798247028146659804, −5.56978337036689272962954494026, −4.85832335020322644656949022876, −2.88065359806123175283508203152, −1.95529027207016422326907919172, 3.80626365783300294641625455862, 4.33423180142113164692504341339, 5.34270592599999670991816940507, 6.46161988222635116305808817171, 7.68084850263016354970411211488, 8.614520039443079548155997227981, 10.25942231786318472738369990331, 11.45450845729989728163747941467, 12.15709455445909574100916041652, 13.46514837534700346438075072783

Graph of the ZZ-function along the critical line