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Properties

Label 2-171-171.121-c1-0-13
Degree $2$
Conductor $171$
Sign $-0.488 + 0.872i$
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  + (1.30 − 2.25i)2-s + (−1.17 + 1.27i)3-s + (−2.38 − 4.13i)4-s + 2.87·5-s + (1.34 + 4.30i)6-s + (−1.80 − 3.12i)7-s − 7.22·8-s + (−0.244 − 2.99i)9-s + (3.74 − 6.48i)10-s + (−0.154 − 0.267i)11-s + (8.07 + 1.81i)12-s + (2.73 + 4.74i)13-s − 9.38·14-s + (−3.37 + 3.66i)15-s + (−4.63 + 8.02i)16-s + (1.60 + 2.78i)17-s + ⋯
L(s)  = 1  + (0.920 − 1.59i)2-s + (−0.677 + 0.735i)3-s + (−1.19 − 2.06i)4-s + 1.28·5-s + (0.548 + 1.75i)6-s + (−0.681 − 1.17i)7-s − 2.55·8-s + (−0.0814 − 0.996i)9-s + (1.18 − 2.05i)10-s + (−0.0466 − 0.0807i)11-s + (2.33 + 0.523i)12-s + (0.759 + 1.31i)13-s − 2.50·14-s + (−0.872 + 0.946i)15-s + (−1.15 + 2.00i)16-s + (0.389 + 0.674i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.488 + 0.872i$
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: \((2,\ 171,\ (\ :1/2),\ -0.488 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778345 - 1.32817i\)
\(L(\frac12)\) \(\approx\) \(0.778345 - 1.32817i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.17 - 1.27i)T \)
19 \( 1 + (-2.06 - 3.83i)T \)
good2 \( 1 + (-1.30 + 2.25i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.87T + 5T^{2} \)
7 \( 1 + (1.80 + 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.154 + 0.267i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.73 - 4.74i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.60 - 2.78i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.598 + 1.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.08T + 29T^{2} \)
31 \( 1 + (0.960 - 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.85T + 37T^{2} \)
41 \( 1 + 8.37T + 41T^{2} \)
43 \( 1 + (-0.880 + 1.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.974T + 47T^{2} \)
53 \( 1 + (2.22 - 3.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.263T + 59T^{2} \)
61 \( 1 - 2.25T + 61T^{2} \)
67 \( 1 + (0.917 + 1.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.72 + 9.91i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.24 + 5.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.50 - 7.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.71 - 2.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.421 - 0.730i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.72 + 6.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.38792365015181966417420960374, −11.34435475415028907750500965074, −10.49148824495436373598232020082, −9.938887254725025192320544850498, −9.242670799677024899356793958554, −6.43991219400224755152094624049, −5.66491073269521825991006743103, −4.31673891053015238128154977693, −3.47064560764950852815086676513, −1.49300890162620344922603698178, 2.86627143980008605040583113844, 5.19967729489138110659717017446, 5.74317909738539487064497225066, 6.35844502688990793226387162891, 7.49736196204761407306671401594, 8.665894109672700881075715344067, 9.812494805806824303940122463486, 11.56522223072228517373648533090, 12.78451206051420788338311027231, 13.13519240229059940933283919574

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