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Properties

Label 2-177-3.2-c4-0-47
Degree $2$
Conductor $177$
Sign $0.0644 - 0.997i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.46i·2-s + (8.98 + 0.580i)3-s + 9.94·4-s + 30.0i·5-s + (−1.42 + 22.0i)6-s + 63.6·7-s + 63.8i·8-s + (80.3 + 10.4i)9-s − 74.0·10-s − 69.6i·11-s + (89.3 + 5.77i)12-s + 35.2·13-s + 156. i·14-s + (−17.4 + 270. i)15-s + 2.04·16-s − 262. i·17-s + ⋯
L(s)  = 1  + 0.615i·2-s + (0.997 + 0.0644i)3-s + 0.621·4-s + 1.20i·5-s + (−0.0396 + 0.613i)6-s + 1.29·7-s + 0.997i·8-s + (0.991 + 0.128i)9-s − 0.740·10-s − 0.575i·11-s + (0.620 + 0.0400i)12-s + 0.208·13-s + 0.799i·14-s + (−0.0776 + 1.20i)15-s + 0.00798·16-s − 0.908i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.0644 - 0.997i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.0644 - 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.592801371\)
\(L(\frac12)\) \(\approx\) \(3.592801371\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.98 - 0.580i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 - 2.46iT - 16T^{2} \)
5 \( 1 - 30.0iT - 625T^{2} \)
7 \( 1 - 63.6T + 2.40e3T^{2} \)
11 \( 1 + 69.6iT - 1.46e4T^{2} \)
13 \( 1 - 35.2T + 2.85e4T^{2} \)
17 \( 1 + 262. iT - 8.35e4T^{2} \)
19 \( 1 + 709.T + 1.30e5T^{2} \)
23 \( 1 + 450. iT - 2.79e5T^{2} \)
29 \( 1 + 125. iT - 7.07e5T^{2} \)
31 \( 1 - 431.T + 9.23e5T^{2} \)
37 \( 1 + 603.T + 1.87e6T^{2} \)
41 \( 1 + 2.85e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.11e3T + 3.41e6T^{2} \)
47 \( 1 - 3.46e3iT - 4.87e6T^{2} \)
53 \( 1 - 5.41e3iT - 7.89e6T^{2} \)
61 \( 1 + 2.38e3T + 1.38e7T^{2} \)
67 \( 1 + 4.38e3T + 2.01e7T^{2} \)
71 \( 1 + 4.29e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.77e3T + 2.83e7T^{2} \)
79 \( 1 - 8.23e3T + 3.89e7T^{2} \)
83 \( 1 + 8.47e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.82e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.00e4T + 8.85e7T^{2} \)
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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

−12.16285092794651419930122863385, −10.88246136107545421420602443645, −10.63053140981286363474055040360, −8.809851989767175277833812121643, −8.034686320222467024263996417064, −7.15771413802510927158886384240, −6.23193839502489007743370094391, −4.57132885416025982304406264655, −2.93895132950891959866615052429, −1.97554302895872767542685383009, 1.40566269015390545896918827456, 2.04512876884370644232256372930, 3.84961344329549150004225139346, 4.86030163459032852679450613452, 6.66817012195639612419270816567, 8.065847705617780257430952923071, 8.538130581436125167161223975771, 9.805947344257275315253087866749, 10.79262043155219376769726548441, 11.89457606644507962541645152244

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