Properties

Label 2-177-59.3-c1-0-3
Degree $2$
Conductor $177$
Sign $0.962 + 0.269i$
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  + (−1.72 + 0.799i)2-s + (−0.947 − 0.319i)3-s + (1.05 − 1.23i)4-s + (−1.39 − 0.837i)5-s + (1.89 − 0.205i)6-s + (−0.142 + 2.62i)7-s + (0.191 − 0.688i)8-s + (0.796 + 0.605i)9-s + (3.07 + 0.334i)10-s + (0.854 − 5.21i)11-s + (−1.39 + 0.837i)12-s + (4.55 − 3.46i)13-s + (−1.85 − 4.64i)14-s + (1.05 + 1.23i)15-s + (0.745 + 4.54i)16-s + (0.344 + 6.35i)17-s + ⋯
L(s)  = 1  + (−1.22 + 0.565i)2-s + (−0.547 − 0.184i)3-s + (0.525 − 0.619i)4-s + (−0.622 − 0.374i)5-s + (0.772 − 0.0840i)6-s + (−0.0537 + 0.991i)7-s + (0.0675 − 0.243i)8-s + (0.265 + 0.201i)9-s + (0.971 + 0.105i)10-s + (0.257 − 1.57i)11-s + (−0.401 + 0.241i)12-s + (1.26 − 0.960i)13-s + (−0.495 − 1.24i)14-s + (0.271 + 0.319i)15-s + (0.186 + 1.13i)16-s + (0.0835 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.962 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465833 - 0.0639617i\)
\(L(\frac12)\) \(\approx\) \(0.465833 - 0.0639617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.947 + 0.319i)T \)
59 \( 1 + (7.59 + 1.13i)T \)
good2 \( 1 + (1.72 - 0.799i)T + (1.29 - 1.52i)T^{2} \)
5 \( 1 + (1.39 + 0.837i)T + (2.34 + 4.41i)T^{2} \)
7 \( 1 + (0.142 - 2.62i)T + (-6.95 - 0.756i)T^{2} \)
11 \( 1 + (-0.854 + 5.21i)T + (-10.4 - 3.51i)T^{2} \)
13 \( 1 + (-4.55 + 3.46i)T + (3.47 - 12.5i)T^{2} \)
17 \( 1 + (-0.344 - 6.35i)T + (-16.9 + 1.83i)T^{2} \)
19 \( 1 + (-4.06 + 3.85i)T + (1.02 - 18.9i)T^{2} \)
23 \( 1 + (-6.63 + 1.46i)T + (20.8 - 9.65i)T^{2} \)
29 \( 1 + (0.602 + 0.278i)T + (18.7 + 22.1i)T^{2} \)
31 \( 1 + (7.43 + 7.04i)T + (1.67 + 30.9i)T^{2} \)
37 \( 1 + (-0.215 - 0.775i)T + (-31.7 + 19.0i)T^{2} \)
41 \( 1 + (-1.27 - 0.280i)T + (37.2 + 17.2i)T^{2} \)
43 \( 1 + (-0.229 - 1.39i)T + (-40.7 + 13.7i)T^{2} \)
47 \( 1 + (-0.00247 + 0.00148i)T + (22.0 - 41.5i)T^{2} \)
53 \( 1 + (3.54 - 0.385i)T + (51.7 - 11.3i)T^{2} \)
61 \( 1 + (-9.21 + 4.26i)T + (39.4 - 46.4i)T^{2} \)
67 \( 1 + (-0.134 + 0.485i)T + (-57.4 - 34.5i)T^{2} \)
71 \( 1 + (-4.07 + 2.45i)T + (33.2 - 62.7i)T^{2} \)
73 \( 1 + (-2.65 - 6.65i)T + (-52.9 + 50.2i)T^{2} \)
79 \( 1 + (-1.81 + 0.612i)T + (62.8 - 47.8i)T^{2} \)
83 \( 1 + (-3.11 + 4.59i)T + (-30.7 - 77.1i)T^{2} \)
89 \( 1 + (-13.4 - 6.24i)T + (57.6 + 67.8i)T^{2} \)
97 \( 1 + (3.81 - 9.58i)T + (-70.4 - 66.7i)T^{2} \)
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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

−12.64055219307459360401957627263, −11.30231510162375585750292585062, −10.78483318449930326496822313273, −9.228531780872119653359187606185, −8.512246119174157965037617798927, −7.86639557314480251625826709281, −6.35998276805642424619287542049, −5.61364939332642566280073759211, −3.59358304313851428051241070767, −0.829312669333990045414705205919, 1.36181163817429330632851859087, 3.61945876798981332801523833053, 5.00553165854392252270719798777, 7.07823001012216016534483432654, 7.41895877397714909012206039469, 9.086170944795550032012056032699, 9.751502851102350858260677040657, 10.80409546956463130833003870615, 11.36692022649816010167184234525, 12.21447695378561172457990592040

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