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Properties

Label 2-171-19.12-c0-0-0
Degree $2$
Conductor $171$
Sign $0.910 - 0.412i$
Analytic cond. $0.0853401$
Root an. cond. $0.292130$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)4-s + 7-s + (−1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s − 19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s + 1.73i·31-s − 1.73i·37-s + (0.5 + 0.866i)43-s + (1.5 − 0.866i)52-s + (−0.5 + 0.866i)61-s + 0.999·64-s + (1.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + 7-s + (−1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s − 19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s + 1.73i·31-s − 1.73i·37-s + (0.5 + 0.866i)43-s + (1.5 − 0.866i)52-s + (−0.5 + 0.866i)61-s + 0.999·64-s + (1.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(0.0853401\)
Root analytic conductor: \(0.292130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :0),\ 0.910 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6304655160\)
\(L(\frac12)\) \(\approx\) \(0.6304655160\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−12.73573705344168905994537873131, −12.38075014513968022941096384101, −11.15249335626099394609718975711, −10.08883893701692750943633178570, −8.795959328665294891122447836875, −8.011942648320657675990551982060, −7.07924459627109291426730101132, −5.24356164546940634662966272452, −4.31033340971662668399967222435, −2.63790986835314549134060667019, 1.99711642451795237760160624628, 4.39128100654410986731420757842, 5.16022945266509770452414471560, 6.54876716174985039446258611077, 7.84292811375560711117310874498, 9.028213104045002098422859164893, 9.884622558268598699539722488447, 10.94297885429766782577005136686, 11.81911774978474639537295195387, 13.04513543539377093871444358350

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