Properties

Degree $2$
Conductor $177$
Sign $0.886 + 0.462i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.46·2-s + (1.16 − 1.28i)3-s + 0.139·4-s + 0.594i·5-s + (1.69 − 1.87i)6-s + 1.86·7-s − 2.72·8-s + (−0.300 − 2.98i)9-s + 0.869i·10-s − 0.676·11-s + (0.161 − 0.178i)12-s + 5.37i·13-s + 2.72·14-s + (0.763 + 0.690i)15-s − 4.25·16-s − 1.70i·17-s + ⋯
L(s)  = 1  + 1.03·2-s + (0.670 − 0.741i)3-s + 0.0695·4-s + 0.265i·5-s + (0.693 − 0.767i)6-s + 0.703·7-s − 0.962·8-s + (−0.100 − 0.994i)9-s + 0.274i·10-s − 0.204·11-s + (0.0466 − 0.0516i)12-s + 1.49i·13-s + 0.727·14-s + (0.197 + 0.178i)15-s − 1.06·16-s − 0.412i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.886 + 0.462i$
Motivic weight: \(1\)
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98694 - 0.486685i\)
\(L(\frac12)\) \(\approx\) \(1.98694 - 0.486685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.16 + 1.28i)T \)
59 \( 1 + (1.93 + 7.43i)T \)
good2 \( 1 - 1.46T + 2T^{2} \)
5 \( 1 - 0.594iT - 5T^{2} \)
7 \( 1 - 1.86T + 7T^{2} \)
11 \( 1 + 0.676T + 11T^{2} \)
13 \( 1 - 5.37iT - 13T^{2} \)
17 \( 1 + 1.70iT - 17T^{2} \)
19 \( 1 + 2.25T + 19T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 - 6.24iT - 29T^{2} \)
31 \( 1 - 2.48iT - 31T^{2} \)
37 \( 1 + 7.86iT - 37T^{2} \)
41 \( 1 - 2.29iT - 41T^{2} \)
43 \( 1 + 7.11iT - 43T^{2} \)
47 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 - 5.73iT - 53T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 + 5.37iT - 67T^{2} \)
71 \( 1 + 5.92iT - 71T^{2} \)
73 \( 1 - 8.26iT - 73T^{2} \)
79 \( 1 - 8.29T + 79T^{2} \)
83 \( 1 + 0.0941T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 17.8iT - 97T^{2} \)
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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

−12.66501087326802583380490259073, −12.06668924830143292318280030249, −10.99225202763936667995701984803, −9.317941350073109270935357392066, −8.563159481649676321505310033087, −7.23797465014828957176202317447, −6.27137909571888206432802433380, −4.83725319731971049382623901596, −3.65602620021831977319715360409, −2.16529180372202488921351917247, 2.71795414198315760237398371845, 4.02562780776801685586261878273, 4.93177564994334675826330912070, 5.91060337434006145927461493305, 7.925881373702258696196200358006, 8.579633172261384297642714163369, 9.844945439052663340477506564780, 10.78445872871378092211578291334, 12.01945680718291286332415295071, 13.07694694704564355389235527751

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