Properties

Label 2-177-177.77-c1-0-0
Degree $2$
Conductor $177$
Sign $0.727 + 0.686i$
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  + (−1.88 − 1.78i)2-s + (−1.60 + 0.658i)3-s + (0.257 + 4.75i)4-s + (−3.64 + 0.598i)5-s + (4.20 + 1.62i)6-s + (−0.920 − 1.73i)7-s + (4.64 − 5.47i)8-s + (2.13 − 2.10i)9-s + (7.95 + 5.39i)10-s + (5.58 + 0.607i)11-s + (−3.54 − 7.44i)12-s + (0.293 + 0.634i)13-s + (−1.36 + 4.92i)14-s + (5.45 − 3.36i)15-s + (−9.09 + 0.988i)16-s + (0.992 + 0.526i)17-s + ⋯
L(s)  = 1  + (−1.33 − 1.26i)2-s + (−0.925 + 0.379i)3-s + (0.128 + 2.37i)4-s + (−1.63 + 0.267i)5-s + (1.71 + 0.662i)6-s + (−0.347 − 0.656i)7-s + (1.64 − 1.93i)8-s + (0.711 − 0.702i)9-s + (2.51 + 1.70i)10-s + (1.68 + 0.183i)11-s + (−1.02 − 2.14i)12-s + (0.0813 + 0.175i)13-s + (−0.365 + 1.31i)14-s + (1.40 − 0.867i)15-s + (−2.27 + 0.247i)16-s + (0.240 + 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.727 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301116 - 0.119700i\)
\(L(\frac12)\) \(\approx\) \(0.301116 - 0.119700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 - 0.658i)T \)
59 \( 1 + (7.65 - 0.625i)T \)
good2 \( 1 + (1.88 + 1.78i)T + (0.108 + 1.99i)T^{2} \)
5 \( 1 + (3.64 - 0.598i)T + (4.73 - 1.59i)T^{2} \)
7 \( 1 + (0.920 + 1.73i)T + (-3.92 + 5.79i)T^{2} \)
11 \( 1 + (-5.58 - 0.607i)T + (10.7 + 2.36i)T^{2} \)
13 \( 1 + (-0.293 - 0.634i)T + (-8.41 + 9.90i)T^{2} \)
17 \( 1 + (-0.992 - 0.526i)T + (9.54 + 14.0i)T^{2} \)
19 \( 1 + (-0.211 - 0.127i)T + (8.89 + 16.7i)T^{2} \)
23 \( 1 + (1.25 - 3.14i)T + (-16.6 - 15.8i)T^{2} \)
29 \( 1 + (-3.31 - 3.49i)T + (-1.57 + 28.9i)T^{2} \)
31 \( 1 + (0.316 + 0.526i)T + (-14.5 + 27.3i)T^{2} \)
37 \( 1 + (-3.47 + 2.95i)T + (5.98 - 36.5i)T^{2} \)
41 \( 1 + (-6.17 + 2.45i)T + (29.7 - 28.1i)T^{2} \)
43 \( 1 + (-0.785 - 7.22i)T + (-41.9 + 9.24i)T^{2} \)
47 \( 1 + (-0.805 + 4.91i)T + (-44.5 - 15.0i)T^{2} \)
53 \( 1 + (-9.97 + 6.76i)T + (19.6 - 49.2i)T^{2} \)
61 \( 1 + (-0.716 + 0.756i)T + (-3.30 - 60.9i)T^{2} \)
67 \( 1 + (-12.3 - 10.4i)T + (10.8 + 66.1i)T^{2} \)
71 \( 1 + (1.54 + 0.252i)T + (67.2 + 22.6i)T^{2} \)
73 \( 1 + (-0.828 - 0.230i)T + (62.5 + 37.6i)T^{2} \)
79 \( 1 + (5.59 - 1.23i)T + (71.6 - 33.1i)T^{2} \)
83 \( 1 + (-3.51 + 2.67i)T + (22.2 - 79.9i)T^{2} \)
89 \( 1 + (-0.370 + 0.351i)T + (4.81 - 88.8i)T^{2} \)
97 \( 1 + (-6.33 + 1.75i)T + (83.1 - 50.0i)T^{2} \)
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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

−11.87718402152808332715907644820, −11.55706156293695234374049086933, −10.73127961578771930235400953169, −9.805974732793306008350305405435, −8.845463026989213667319029102242, −7.56501543716783085928541746095, −6.76485683794368074884429603066, −4.10798712143600667323583030481, −3.62712181531140314280421017358, −0.911037563008825988999660000846, 0.808388154884904260393180493498, 4.38128671823737449415552305361, 5.89682815701176751294711889458, 6.71917174069247230472305812217, 7.65024508944598110598871372074, 8.521075447404108145533065059847, 9.458245588466569151270422194100, 10.78670385393404335321760162872, 11.73772749954766362606772389805, 12.37466107973945453099192378152

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