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Properties

Label 2-171-171.106-c1-0-7
Degree 22
Conductor 171171
Sign 0.185+0.982i0.185 + 0.982i
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  + (−1.23 − 2.13i)2-s + (1.62 + 0.605i)3-s + (−2.04 + 3.54i)4-s + 1.91·5-s + (−0.709 − 4.21i)6-s + (−0.324 + 0.562i)7-s + 5.16·8-s + (2.26 + 1.96i)9-s + (−2.36 − 4.09i)10-s + (2.93 − 5.07i)11-s + (−5.46 + 4.51i)12-s + (0.327 − 0.567i)13-s + 1.60·14-s + (3.10 + 1.15i)15-s + (−2.27 − 3.94i)16-s + (−1.93 + 3.35i)17-s + ⋯
L(s)  = 1  + (−0.872 − 1.51i)2-s + (0.936 + 0.349i)3-s + (−1.02 + 1.77i)4-s + 0.856·5-s + (−0.289 − 1.72i)6-s + (−0.122 + 0.212i)7-s + 1.82·8-s + (0.755 + 0.654i)9-s + (−0.747 − 1.29i)10-s + (0.884 − 1.53i)11-s + (−1.57 + 1.30i)12-s + (0.0908 − 0.157i)13-s + 0.428·14-s + (0.802 + 0.299i)15-s + (−0.569 − 0.986i)16-s + (−0.469 + 0.813i)17-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((0.185+0.982i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((0.185+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: 0.185+0.982i0.185 + 0.982i
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), 0.185+0.982i)(2,\ 171,\ (\ :1/2),\ 0.185 + 0.982i)

Particular Values

L(1)L(1) ≈\approx 0.831057−0.688829i0.831057 - 0.688829i
L(12)L(\frac12) ≈\approx 0.831057−0.688829i0.831057 - 0.688829i
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+(−1.62−0.605i)T 1 + (-1.62 - 0.605i)T
19 1+(4.28+0.802i)T 1 + (4.28 + 0.802i)T
good2 1+(1.23+2.13i)T+(−1+1.73i)T2 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2}
5 1−1.91T+5T2 1 - 1.91T + 5T^{2}
7 1+(0.324−0.562i)T+(−3.5−6.06i)T2 1 + (0.324 - 0.562i)T + (-3.5 - 6.06i)T^{2}
11 1+(−2.93+5.07i)T+(−5.5−9.52i)T2 1 + (-2.93 + 5.07i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.327+0.567i)T+(−6.5−11.2i)T2 1 + (-0.327 + 0.567i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.93−3.35i)T+(−8.5−14.7i)T2 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2}
23 1+(−0.961+1.66i)T+(−11.5−19.9i)T2 1 + (-0.961 + 1.66i)T + (-11.5 - 19.9i)T^{2}
29 1+6.53T+29T2 1 + 6.53T + 29T^{2}
31 1+(−1.54−2.67i)T+(−15.5+26.8i)T2 1 + (-1.54 - 2.67i)T + (-15.5 + 26.8i)T^{2}
37 1−2.23T+37T2 1 - 2.23T + 37T^{2}
41 1+6.96T+41T2 1 + 6.96T + 41T^{2}
43 1+(−4.46−7.74i)T+(−21.5+37.2i)T2 1 + (-4.46 - 7.74i)T + (-21.5 + 37.2i)T^{2}
47 1−11.5T+47T2 1 - 11.5T + 47T^{2}
53 1+(6.35+11.0i)T+(−26.5+45.8i)T2 1 + (6.35 + 11.0i)T + (-26.5 + 45.8i)T^{2}
59 1+14.3T+59T2 1 + 14.3T + 59T^{2}
61 1+10.3T+61T2 1 + 10.3T + 61T^{2}
67 1+(0.381−0.661i)T+(−33.5−58.0i)T2 1 + (0.381 - 0.661i)T + (-33.5 - 58.0i)T^{2}
71 1+(0.299−0.519i)T+(−35.5−61.4i)T2 1 + (0.299 - 0.519i)T + (-35.5 - 61.4i)T^{2}
73 1+(−1.75+3.04i)T+(−36.5−63.2i)T2 1 + (-1.75 + 3.04i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.13−3.69i)T+(−39.5+68.4i)T2 1 + (-2.13 - 3.69i)T + (-39.5 + 68.4i)T^{2}
83 1+(3.29−5.71i)T+(−41.5−71.8i)T2 1 + (3.29 - 5.71i)T + (-41.5 - 71.8i)T^{2}
89 1+(−2.41−4.19i)T+(−44.5+77.0i)T2 1 + (-2.41 - 4.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.19+2.06i)T+(−48.5+84.0i)T2 1 + (1.19 + 2.06i)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

−12.51591914886383858997629471552, −11.11546447082412116239057201466, −10.56772799699368838915763253669, −9.422466710067938006451847679038, −8.923725031885829404162684082718, −8.115737977828802583096742020478, −6.16514109002018044078096039526, −4.07212554775304170138788479869, −2.96959485434975587517884855823, −1.71228902011864065981846341090, 1.88000843457016246218744997059, 4.39297966952684168607043247205, 6.03666164506989404516090252639, 6.98769665962023641389229122204, 7.61268050746769297682594071629, 9.083139576180760988120543481853, 9.325887357725773294935539976769, 10.30261371659872136522659335993, 12.26855229063827268401150450304, 13.51671888541406530862662996935

Graph of the ZZ-function along the critical line