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Properties

Label 2-171-171.121-c1-0-5
Degree $2$
Conductor $171$
Sign $0.185 - 0.982i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.185 - 0.982i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831057 + 0.688829i\)
\(L(\frac12)\) \(\approx\) \(0.831057 + 0.688829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 + 0.605i)T \)
19 \( 1 + (4.28 - 0.802i)T \)
good2 \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.91T + 5T^{2} \)
7 \( 1 + (0.324 + 0.562i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.93 - 5.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.327 - 0.567i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 + 3.35i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.961 - 1.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.53T + 29T^{2} \)
31 \( 1 + (-1.54 + 2.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.23T + 37T^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 + (-4.46 + 7.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + (6.35 - 11.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + (0.381 + 0.661i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.299 + 0.519i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.75 - 3.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.13 + 3.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.29 + 5.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.41 + 4.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.19 - 2.06i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line