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Properties

Label 2-171-171.106-c1-0-8
Degree 22
Conductor 171171
Sign −0.322−0.946i-0.322 - 0.946i
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  + (1.30 + 2.26i)2-s + (1.63 − 0.581i)3-s + (−2.40 + 4.16i)4-s − 2.00·5-s + (3.44 + 2.92i)6-s + (0.257 − 0.445i)7-s − 7.34·8-s + (2.32 − 1.89i)9-s + (−2.61 − 4.52i)10-s + (2.04 − 3.54i)11-s + (−1.50 + 8.20i)12-s + (−1.85 + 3.20i)13-s + 1.34·14-s + (−3.26 + 1.16i)15-s + (−4.77 − 8.26i)16-s + (3.60 − 6.24i)17-s + ⋯
L(s)  = 1  + (0.922 + 1.59i)2-s + (0.941 − 0.335i)3-s + (−1.20 + 2.08i)4-s − 0.895·5-s + (1.40 + 1.19i)6-s + (0.0971 − 0.168i)7-s − 2.59·8-s + (0.774 − 0.632i)9-s + (−0.826 − 1.43i)10-s + (0.617 − 1.06i)11-s + (−0.433 + 2.36i)12-s + (−0.513 + 0.889i)13-s + 0.358·14-s + (−0.843 + 0.300i)15-s + (−1.19 − 2.06i)16-s + (0.874 − 1.51i)17-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((−0.322−0.946i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((−0.322−0.946i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: −0.322−0.946i-0.322 - 0.946i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ171(106,⋅)\chi_{171} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), −0.322−0.946i)(2,\ 171,\ (\ :1/2),\ -0.322 - 0.946i)

Particular Values

L(1)L(1) ≈\approx 1.13024+1.57870i1.13024 + 1.57870i
L(12)L(\frac12) ≈\approx 1.13024+1.57870i1.13024 + 1.57870i
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+(−1.63+0.581i)T 1 + (-1.63 + 0.581i)T
19 1+(−0.559−4.32i)T 1 + (-0.559 - 4.32i)T
good2 1+(−1.30−2.26i)T+(−1+1.73i)T2 1 + (-1.30 - 2.26i)T + (-1 + 1.73i)T^{2}
5 1+2.00T+5T2 1 + 2.00T + 5T^{2}
7 1+(−0.257+0.445i)T+(−3.5−6.06i)T2 1 + (-0.257 + 0.445i)T + (-3.5 - 6.06i)T^{2}
11 1+(−2.04+3.54i)T+(−5.5−9.52i)T2 1 + (-2.04 + 3.54i)T + (-5.5 - 9.52i)T^{2}
13 1+(1.85−3.20i)T+(−6.5−11.2i)T2 1 + (1.85 - 3.20i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.60+6.24i)T+(−8.5−14.7i)T2 1 + (-3.60 + 6.24i)T + (-8.5 - 14.7i)T^{2}
23 1+(0.174−0.301i)T+(−11.5−19.9i)T2 1 + (0.174 - 0.301i)T + (-11.5 - 19.9i)T^{2}
29 1+7.54T+29T2 1 + 7.54T + 29T^{2}
31 1+(0.773+1.33i)T+(−15.5+26.8i)T2 1 + (0.773 + 1.33i)T + (-15.5 + 26.8i)T^{2}
37 1+6.82T+37T2 1 + 6.82T + 37T^{2}
41 1+2.92T+41T2 1 + 2.92T + 41T^{2}
43 1+(0.200+0.347i)T+(−21.5+37.2i)T2 1 + (0.200 + 0.347i)T + (-21.5 + 37.2i)T^{2}
47 1−6.16T+47T2 1 - 6.16T + 47T^{2}
53 1+(−2.35−4.08i)T+(−26.5+45.8i)T2 1 + (-2.35 - 4.08i)T + (-26.5 + 45.8i)T^{2}
59 1+4.30T+59T2 1 + 4.30T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1+(0.480−0.831i)T+(−33.5−58.0i)T2 1 + (0.480 - 0.831i)T + (-33.5 - 58.0i)T^{2}
71 1+(3.26−5.66i)T+(−35.5−61.4i)T2 1 + (3.26 - 5.66i)T + (-35.5 - 61.4i)T^{2}
73 1+(−1.31+2.28i)T+(−36.5−63.2i)T2 1 + (-1.31 + 2.28i)T + (-36.5 - 63.2i)T^{2}
79 1+(3.55+6.16i)T+(−39.5+68.4i)T2 1 + (3.55 + 6.16i)T + (-39.5 + 68.4i)T^{2}
83 1+(8.37−14.5i)T+(−41.5−71.8i)T2 1 + (8.37 - 14.5i)T + (-41.5 - 71.8i)T^{2}
89 1+(5.10+8.84i)T+(−44.5+77.0i)T2 1 + (5.10 + 8.84i)T + (-44.5 + 77.0i)T^{2}
97 1+(−9.64−16.7i)T+(−48.5+84.0i)T2 1 + (-9.64 - 16.7i)T + (-48.5 + 84.0i)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

−13.63598290166328317982194770479, −12.33388863133746939144229819840, −11.68723526206542324716659938669, −9.445474123074456407349670619585, −8.513637892063063148817682054343, −7.56082407912426544499882075301, −7.06368290292929368560309996067, −5.67292024831985270013248481183, −4.17145746361551894541457160633, −3.39232090812837591909328876040, 1.94269311228105392462784841361, 3.39495402424805896350554164810, 4.13488854566990339812366628071, 5.28726544753036826555327950284, 7.41497213647203814424687766371, 8.721567436606042291032963778310, 9.840344546824306101598184633244, 10.51561076345526183009594925748, 11.66368162646693125200332750352, 12.50865762800835943760350577195

Graph of the ZZ-function along the critical line