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Properties

Label 2-171-57.56-c1-0-4
Degree $2$
Conductor $171$
Sign $0.577 + 0.816i$
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  − 2·4-s − 2.37i·5-s + 4.35·7-s − 6.61i·11-s + 4·16-s + 1.86i·17-s − 4.35·19-s + 4.75i·20-s + 8.99i·23-s − 0.641·25-s − 8.71·28-s − 10.3i·35-s − 43-s + 13.2i·44-s + 6.11i·47-s + ⋯
L(s)  = 1  − 4-s − 1.06i·5-s + 1.64·7-s − 1.99i·11-s + 16-s + 0.452i·17-s − 1.00·19-s + 1.06i·20-s + 1.87i·23-s − 0.128·25-s − 1.64·28-s − 1.74i·35-s − 0.152·43-s + 1.99i·44-s + 0.891i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931848 - 0.482360i\)
\(L(\frac12)\) \(\approx\) \(0.931848 - 0.482360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + 4.35T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + 2.37iT - 5T^{2} \)
7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 + 6.61iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 1.86iT - 17T^{2} \)
23 \( 1 - 8.99iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 6.11iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 4.35T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.76832591629747791780056268042, −11.56702058787996924257041615799, −10.78807663601031237718037695283, −9.245015387463118943826530040374, −8.433774734769499447297411708611, −8.022828193466520071020103123035, −5.74748405417098613937629589742, −4.99029040060361356124191245892, −3.84244602729586642983218618364, −1.19761556299323890853626080839, 2.19220898156498664547707092403, 4.30063438461480201546142650044, 4.96946250265873615426572728138, 6.76307186717429580077078497625, 7.79174925641288207389001775538, 8.776685635403250871995095011550, 10.08470214758574881223163686046, 10.74304417909708818750143435461, 11.97359450962457679771706702162, 12.87535129756270034798468957719

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