Properties

Label 2-177-3.2-c2-0-36
Degree $2$
Conductor $177$
Sign $-0.575 - 0.818i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.77i·2-s + (2.45 − 1.72i)3-s − 10.2·4-s − 1.62i·5-s + (−6.52 − 9.27i)6-s − 7.78·7-s + 23.7i·8-s + (3.04 − 8.46i)9-s − 6.15·10-s + 2.72i·11-s + (−25.2 + 17.7i)12-s + 9.61·13-s + 29.4i·14-s + (−2.80 − 3.99i)15-s + 48.6·16-s − 27.3i·17-s + ⋯
L(s)  = 1  − 1.88i·2-s + (0.818 − 0.575i)3-s − 2.57·4-s − 0.325i·5-s + (−1.08 − 1.54i)6-s − 1.11·7-s + 2.97i·8-s + (0.338 − 0.940i)9-s − 0.615·10-s + 0.248i·11-s + (−2.10 + 1.47i)12-s + 0.739·13-s + 2.10i·14-s + (−0.187 − 0.266i)15-s + 3.04·16-s − 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.575 - 0.818i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.575 - 0.818i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.577749 + 1.11235i\)
\(L(\frac12)\) \(\approx\) \(0.577749 + 1.11235i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.45 + 1.72i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 3.77iT - 4T^{2} \)
5 \( 1 + 1.62iT - 25T^{2} \)
7 \( 1 + 7.78T + 49T^{2} \)
11 \( 1 - 2.72iT - 121T^{2} \)
13 \( 1 - 9.61T + 169T^{2} \)
17 \( 1 + 27.3iT - 289T^{2} \)
19 \( 1 + 27.7T + 361T^{2} \)
23 \( 1 + 3.53iT - 529T^{2} \)
29 \( 1 + 6.30iT - 841T^{2} \)
31 \( 1 - 39.9T + 961T^{2} \)
37 \( 1 - 44.2T + 1.36e3T^{2} \)
41 \( 1 + 50.1iT - 1.68e3T^{2} \)
43 \( 1 + 53.9T + 1.84e3T^{2} \)
47 \( 1 - 47.0iT - 2.20e3T^{2} \)
53 \( 1 + 9.97iT - 2.80e3T^{2} \)
61 \( 1 + 71.5T + 3.72e3T^{2} \)
67 \( 1 - 86.8T + 4.48e3T^{2} \)
71 \( 1 + 119. iT - 5.04e3T^{2} \)
73 \( 1 - 16.4T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 - 125. iT - 6.88e3T^{2} \)
89 \( 1 - 137. iT - 7.92e3T^{2} \)
97 \( 1 - 0.0640T + 9.40e3T^{2} \)
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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

−12.06163689449325587717725331978, −10.87453532093562527190949297259, −9.746898104967779970808158337094, −9.155710818625517462201212403396, −8.234246454950580632689928692015, −6.52901162948535750108760593428, −4.56291520717395643890352281652, −3.34065175021626594850917219128, −2.38202084120453341056919212713, −0.71348416701859677728474483483, 3.43990278632188987144493992578, 4.49188397446461942470524143748, 6.08926360423633249167102409264, 6.68814743361468796804057592407, 8.158303096236679925683259165208, 8.627015357334832068292630471145, 9.727565465766888087350147401908, 10.58962573817458119216673718558, 12.97056197330445969773512413806, 13.27373058825400426359674671298

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