Properties

Label 2-177-59.4-c3-0-14
Degree $2$
Conductor $177$
Sign $0.517 + 0.855i$
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  + (0.115 + 0.0125i)2-s + (−1.94 + 2.28i)3-s + (−7.79 − 1.71i)4-s + (−3.58 − 2.72i)5-s + (−0.252 + 0.238i)6-s + (−13.1 + 32.8i)7-s + (−1.75 − 0.590i)8-s + (−1.45 − 8.88i)9-s + (−0.378 − 0.358i)10-s + (54.4 − 25.2i)11-s + (19.0 − 14.4i)12-s + (5.23 − 31.9i)13-s + (−1.91 + 3.61i)14-s + (13.2 − 2.90i)15-s + (57.7 + 26.7i)16-s + (−43.9 − 110. i)17-s + ⋯
L(s)  = 1  + (0.0406 + 0.00442i)2-s + (−0.373 + 0.440i)3-s + (−0.974 − 0.214i)4-s + (−0.320 − 0.243i)5-s + (−0.0171 + 0.0162i)6-s + (−0.707 + 1.77i)7-s + (−0.0774 − 0.0261i)8-s + (−0.0539 − 0.328i)9-s + (−0.0119 − 0.0113i)10-s + (1.49 − 0.691i)11-s + (0.458 − 0.348i)12-s + (0.111 − 0.680i)13-s + (−0.0366 + 0.0691i)14-s + (0.227 − 0.0500i)15-s + (0.903 + 0.417i)16-s + (−0.626 − 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.517 + 0.855i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 0.517 + 0.855i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.666522 - 0.375668i\)
\(L(\frac12)\) \(\approx\) \(0.666522 - 0.375668i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.94 - 2.28i)T \)
59 \( 1 + (-453. + 3.65i)T \)
good2 \( 1 + (-0.115 - 0.0125i)T + (7.81 + 1.71i)T^{2} \)
5 \( 1 + (3.58 + 2.72i)T + (33.4 + 120. i)T^{2} \)
7 \( 1 + (13.1 - 32.8i)T + (-249. - 235. i)T^{2} \)
11 \( 1 + (-54.4 + 25.2i)T + (861. - 1.01e3i)T^{2} \)
13 \( 1 + (-5.23 + 31.9i)T + (-2.08e3 - 701. i)T^{2} \)
17 \( 1 + (43.9 + 110. i)T + (-3.56e3 + 3.37e3i)T^{2} \)
19 \( 1 + (24.7 + 36.4i)T + (-2.53e3 + 6.37e3i)T^{2} \)
23 \( 1 + (3.13 - 57.8i)T + (-1.20e4 - 1.31e3i)T^{2} \)
29 \( 1 + (-136. + 14.8i)T + (2.38e4 - 5.24e3i)T^{2} \)
31 \( 1 + (55.2 - 81.4i)T + (-1.10e4 - 2.76e4i)T^{2} \)
37 \( 1 + (-366. + 123. i)T + (4.03e4 - 3.06e4i)T^{2} \)
41 \( 1 + (1.09 + 20.2i)T + (-6.85e4 + 7.45e3i)T^{2} \)
43 \( 1 + (-119. - 55.2i)T + (5.14e4 + 6.05e4i)T^{2} \)
47 \( 1 + (297. - 226. i)T + (2.77e4 - 1.00e5i)T^{2} \)
53 \( 1 + (-190. + 180. i)T + (8.06e3 - 1.48e5i)T^{2} \)
61 \( 1 + (156. + 17.0i)T + (2.21e5 + 4.87e4i)T^{2} \)
67 \( 1 + (374. + 126. i)T + (2.39e5 + 1.82e5i)T^{2} \)
71 \( 1 + (-616. + 468. i)T + (9.57e4 - 3.44e5i)T^{2} \)
73 \( 1 + (-186. + 352. i)T + (-2.18e5 - 3.21e5i)T^{2} \)
79 \( 1 + (529. + 623. i)T + (-7.97e4 + 4.86e5i)T^{2} \)
83 \( 1 + (98.8 - 59.4i)T + (2.67e5 - 5.05e5i)T^{2} \)
89 \( 1 + (1.18e3 - 128. i)T + (6.88e5 - 1.51e5i)T^{2} \)
97 \( 1 + (633. + 1.19e3i)T + (-5.12e5 + 7.55e5i)T^{2} \)
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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

−12.04477715461459164231227295969, −11.30761943840435134343606801882, −9.678816291774590749409774008633, −9.153929945717585545222041002204, −8.436786228902483913690614397072, −6.41187227569225495802721769058, −5.56479309063383654530680812315, −4.44945243563390513715777059367, −3.04419524149546522169396641988, −0.44907334658374577413932166878, 1.17573710578897356190966855574, 3.92526330581019738008828680635, 4.22682074138901641090632540000, 6.34165380249518911354398596200, 7.04637917575553224825766127103, 8.222510439729807167825766255488, 9.488502929978440743460803870470, 10.35537601087789449042120893902, 11.45479850537259656262647315235, 12.62582465773021168635925297849

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