Properties

Label 2-177-177.2-c3-0-26
Degree $2$
Conductor $177$
Sign $-0.959 + 0.280i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.214 + 3.95i)2-s + (4.30 + 2.91i)3-s + (−7.64 − 0.831i)4-s + (−4.87 + 14.4i)5-s + (−12.4 + 16.3i)6-s + (8.97 + 13.2i)7-s + (−0.198 + 1.21i)8-s + (9.99 + 25.0i)9-s + (−56.2 − 22.3i)10-s + (49.8 − 10.9i)11-s + (−30.4 − 25.8i)12-s + (52.0 − 44.1i)13-s + (−54.2 + 32.6i)14-s + (−63.1 + 48.0i)15-s + (−64.8 − 14.2i)16-s + (−66.4 − 45.0i)17-s + ⋯
L(s)  = 1  + (−0.0758 + 1.39i)2-s + (0.827 + 0.561i)3-s + (−0.955 − 0.103i)4-s + (−0.436 + 1.29i)5-s + (−0.847 + 1.11i)6-s + (0.484 + 0.714i)7-s + (−0.00878 + 0.0536i)8-s + (0.370 + 0.928i)9-s + (−1.77 − 0.708i)10-s + (1.36 − 0.300i)11-s + (−0.732 − 0.622i)12-s + (1.10 − 0.942i)13-s + (−1.03 + 0.623i)14-s + (−1.08 + 0.826i)15-s + (−1.01 − 0.222i)16-s + (−0.948 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.959 + 0.280i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.959 + 0.280i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.317580 - 2.21520i\)
\(L(\frac12)\) \(\approx\) \(0.317580 - 2.21520i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.30 - 2.91i)T \)
59 \( 1 + (-104. - 440. i)T \)
good2 \( 1 + (0.214 - 3.95i)T + (-7.95 - 0.864i)T^{2} \)
5 \( 1 + (4.87 - 14.4i)T + (-99.5 - 75.6i)T^{2} \)
7 \( 1 + (-8.97 - 13.2i)T + (-126. + 318. i)T^{2} \)
11 \( 1 + (-49.8 + 10.9i)T + (1.20e3 - 558. i)T^{2} \)
13 \( 1 + (-52.0 + 44.1i)T + (355. - 2.16e3i)T^{2} \)
17 \( 1 + (66.4 + 45.0i)T + (1.81e3 + 4.56e3i)T^{2} \)
19 \( 1 + (-22.2 + 41.9i)T + (-3.84e3 - 5.67e3i)T^{2} \)
23 \( 1 + (-80.4 + 76.2i)T + (658. - 1.21e4i)T^{2} \)
29 \( 1 + (265. - 14.3i)T + (2.42e4 - 2.63e3i)T^{2} \)
31 \( 1 + (43.3 - 23.0i)T + (1.67e4 - 2.46e4i)T^{2} \)
37 \( 1 + (-270. + 44.3i)T + (4.80e4 - 1.61e4i)T^{2} \)
41 \( 1 + (-77.4 + 81.8i)T + (-3.73e3 - 6.88e4i)T^{2} \)
43 \( 1 + (-20.5 + 93.4i)T + (-7.21e4 - 3.33e4i)T^{2} \)
47 \( 1 + (-246. + 83.0i)T + (8.26e4 - 6.28e4i)T^{2} \)
53 \( 1 + (42.6 - 17.0i)T + (1.08e5 - 1.02e5i)T^{2} \)
61 \( 1 + (488. + 26.4i)T + (2.25e5 + 2.45e4i)T^{2} \)
67 \( 1 + (-741. - 121. i)T + (2.85e5 + 9.60e4i)T^{2} \)
71 \( 1 + (-270. - 802. i)T + (-2.84e5 + 2.16e5i)T^{2} \)
73 \( 1 + (166. + 276. i)T + (-1.82e5 + 3.43e5i)T^{2} \)
79 \( 1 + (358. + 165. i)T + (3.19e5 + 3.75e5i)T^{2} \)
83 \( 1 + (-190. - 684. i)T + (-4.89e5 + 2.94e5i)T^{2} \)
89 \( 1 + (82.1 + 1.51e3i)T + (-7.00e5 + 7.62e4i)T^{2} \)
97 \( 1 + (469. - 780. i)T + (-4.27e5 - 8.06e5i)T^{2} \)
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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

−13.31041308869890045991810083296, −11.37574013174838957864193826937, −10.92910956021460059862307466350, −9.191305640384447338492769858621, −8.642119931594549045661626930299, −7.54217799677613178834401604536, −6.70458081202577958783141824411, −5.52166710158806286546725287375, −3.99073656741209708366018957359, −2.65554284385426310567638473535, 1.11507633075267798770360391967, 1.69202721161374421812220659265, 3.83845729515153941447160436277, 4.20737378558021793699529165115, 6.56014304780814795566023940621, 7.87943209357720950226197036620, 9.133264942631750100214872746141, 9.284128707943542495975842643450, 11.08025948682590786341534182314, 11.66615327711456381793096496295

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