Properties

Label 2-177-177.11-c1-0-12
Degree $2$
Conductor $177$
Sign $0.870 + 0.491i$
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  + (1.62 − 0.357i)2-s + (−1.66 − 0.474i)3-s + (0.697 − 0.322i)4-s + (2.79 + 0.774i)5-s + (−2.87 − 0.175i)6-s + (2.93 − 2.78i)7-s + (−1.63 + 1.23i)8-s + (2.54 + 1.58i)9-s + (4.81 + 0.260i)10-s + (−1.85 − 2.17i)11-s + (−1.31 + 0.206i)12-s + (−0.767 − 2.27i)13-s + (3.77 − 5.57i)14-s + (−4.28 − 2.61i)15-s + (−3.20 + 3.77i)16-s + (−3.60 + 3.80i)17-s + ⋯
L(s)  = 1  + (1.14 − 0.252i)2-s + (−0.961 − 0.274i)3-s + (0.348 − 0.161i)4-s + (1.24 + 0.346i)5-s + (−1.17 − 0.0716i)6-s + (1.11 − 1.05i)7-s + (−0.576 + 0.438i)8-s + (0.849 + 0.527i)9-s + (1.52 + 0.0825i)10-s + (−0.558 − 0.657i)11-s + (−0.379 + 0.0596i)12-s + (−0.212 − 0.631i)13-s + (1.01 − 1.48i)14-s + (−1.10 − 0.675i)15-s + (−0.800 + 0.942i)16-s + (−0.875 + 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.870 + 0.491i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.870 + 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69169 - 0.444201i\)
\(L(\frac12)\) \(\approx\) \(1.69169 - 0.444201i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 + 0.474i)T \)
59 \( 1 + (-6.61 + 3.90i)T \)
good2 \( 1 + (-1.62 + 0.357i)T + (1.81 - 0.839i)T^{2} \)
5 \( 1 + (-2.79 - 0.774i)T + (4.28 + 2.57i)T^{2} \)
7 \( 1 + (-2.93 + 2.78i)T + (0.378 - 6.98i)T^{2} \)
11 \( 1 + (1.85 + 2.17i)T + (-1.77 + 10.8i)T^{2} \)
13 \( 1 + (0.767 + 2.27i)T + (-10.3 + 7.86i)T^{2} \)
17 \( 1 + (3.60 - 3.80i)T + (-0.920 - 16.9i)T^{2} \)
19 \( 1 + (-2.04 - 5.13i)T + (-13.7 + 13.0i)T^{2} \)
23 \( 1 + (-0.428 + 0.0466i)T + (22.4 - 4.94i)T^{2} \)
29 \( 1 + (1.38 - 6.29i)T + (-26.3 - 12.1i)T^{2} \)
31 \( 1 + (6.53 + 2.60i)T + (22.5 + 21.3i)T^{2} \)
37 \( 1 + (1.66 - 2.19i)T + (-9.89 - 35.6i)T^{2} \)
41 \( 1 + (-0.390 + 3.59i)T + (-40.0 - 8.81i)T^{2} \)
43 \( 1 + (3.52 + 2.99i)T + (6.95 + 42.4i)T^{2} \)
47 \( 1 + (2.39 + 8.63i)T + (-40.2 + 24.2i)T^{2} \)
53 \( 1 + (-9.81 + 0.532i)T + (52.6 - 5.73i)T^{2} \)
61 \( 1 + (-1.43 - 6.52i)T + (-55.3 + 25.6i)T^{2} \)
67 \( 1 + (-8.25 - 10.8i)T + (-17.9 + 64.5i)T^{2} \)
71 \( 1 + (4.00 - 1.11i)T + (60.8 - 36.6i)T^{2} \)
73 \( 1 + (7.57 + 5.13i)T + (27.0 + 67.8i)T^{2} \)
79 \( 1 + (-1.30 - 7.93i)T + (-74.8 + 25.2i)T^{2} \)
83 \( 1 + (1.51 + 2.86i)T + (-46.5 + 68.6i)T^{2} \)
89 \( 1 + (-6.64 - 1.46i)T + (80.7 + 37.3i)T^{2} \)
97 \( 1 + (-2.81 + 1.91i)T + (35.9 - 90.1i)T^{2} \)
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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

−12.93698905572385293240427833778, −11.68935493353885989720521021687, −10.75299218756370124221302972477, −10.25930309511181305992355426278, −8.388847481162479436241875709266, −7.03549351360426038449842747105, −5.71933646788199275180790030266, −5.26568748606511813782173171138, −3.87178196579561808100267363386, −1.89173721759644350823217593687, 2.23248051822758542343435620321, 4.68116251696239290895194233646, 5.08607187873074065302198727761, 5.92305944922108562030830481787, 7.06499076742737247839665100324, 9.100308601773588849837279503916, 9.653300186163879091839550125425, 11.19893379585181053363950641572, 11.90870706863188021663403954573, 12.90797174068771351722301717293

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