Properties

Label 2-177-177.176-c1-0-4
Degree $2$
Conductor $177$
Sign $0.548 - 0.835i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
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  − 1.46·2-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.548 - 0.835i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.548 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746991 + 0.403201i\)
\(L(\frac12)\) \(\approx\) \(0.746991 + 0.403201i\)
\(L(\frac{3}{2})\) 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

−12.97160273906311844282082172769, −11.41174452074998572717206349093, −10.65401409469725502124176784861, −9.566657317995193996788234351183, −8.860679386919625725725583000255, −8.220486455851872928995462600791, −6.99717277657502639984462377315, −4.96340666105419233682169571608, −4.06977282938096059805462890001, −1.90169836458706514322098136145, 1.22245333704650313101855834760, 3.01009819221546976033602332409, 4.91737230872597404630533303827, 6.69370075700139909522987664644, 7.74723413978053675332335987773, 8.375632114285907654383671805369, 9.283157073546609192752580622838, 10.40572089071951141356547765419, 11.32134865304277437693689637908, 12.71901281549934940078551221199

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