Properties

Label 2-177-3.2-c2-0-0
Degree $2$
Conductor $177$
Sign $0.825 - 0.564i$
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  − 3.55i·2-s + (−1.69 − 2.47i)3-s − 8.66·4-s + 6.82i·5-s + (−8.81 + 6.02i)6-s + 0.148·7-s + 16.5i·8-s + (−3.26 + 8.38i)9-s + 24.2·10-s − 1.80i·11-s + (14.6 + 21.4i)12-s − 17.1·13-s − 0.528i·14-s + (16.8 − 11.5i)15-s + 24.3·16-s + 18.3i·17-s + ⋯
L(s)  = 1  − 1.77i·2-s + (−0.564 − 0.825i)3-s − 2.16·4-s + 1.36i·5-s + (−1.46 + 1.00i)6-s + 0.0212·7-s + 2.07i·8-s + (−0.363 + 0.931i)9-s + 2.42·10-s − 0.164i·11-s + (1.22 + 1.78i)12-s − 1.31·13-s − 0.0377i·14-s + (1.12 − 0.770i)15-s + 1.52·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.825 - 0.564i$
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: \((2,\ 177,\ (\ :1),\ 0.825 - 0.564i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.156074 + 0.0482438i\)
\(L(\frac12)\) \(\approx\) \(0.156074 + 0.0482438i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 2.47i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 + 3.55iT - 4T^{2} \)
5 \( 1 - 6.82iT - 25T^{2} \)
7 \( 1 - 0.148T + 49T^{2} \)
11 \( 1 + 1.80iT - 121T^{2} \)
13 \( 1 + 17.1T + 169T^{2} \)
17 \( 1 - 18.3iT - 289T^{2} \)
19 \( 1 + 19.6T + 361T^{2} \)
23 \( 1 + 40.3iT - 529T^{2} \)
29 \( 1 - 33.8iT - 841T^{2} \)
31 \( 1 + 46.0T + 961T^{2} \)
37 \( 1 - 40.6T + 1.36e3T^{2} \)
41 \( 1 + 21.3iT - 1.68e3T^{2} \)
43 \( 1 - 4.07T + 1.84e3T^{2} \)
47 \( 1 - 45.7iT - 2.20e3T^{2} \)
53 \( 1 - 29.4iT - 2.80e3T^{2} \)
61 \( 1 + 28.5T + 3.72e3T^{2} \)
67 \( 1 + 6.55T + 4.48e3T^{2} \)
71 \( 1 - 11.4iT - 5.04e3T^{2} \)
73 \( 1 + 113.T + 5.32e3T^{2} \)
79 \( 1 + 32.0T + 6.24e3T^{2} \)
83 \( 1 + 145. iT - 6.88e3T^{2} \)
89 \( 1 + 148. iT - 7.92e3T^{2} \)
97 \( 1 + 133.T + 9.40e3T^{2} \)
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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

−12.62903163786653058334117844900, −11.42395336607779734250773400175, −10.70366112213110789498307417889, −10.26728997916418017856939954773, −8.740195255925333141688331870062, −7.36348249567530651114886486675, −6.18965433917956431352376145466, −4.54274754168535961174701538815, −2.94155994417372611549768797491, −1.98646063558912228310615912089, 0.10162709168686636136030408094, 4.25334742704471340699668234975, 5.03209314029665607402520409265, 5.71959656671048741346569198210, 7.10204827276295173647934503674, 8.170030417150509097648334872126, 9.350459981208556272812760922671, 9.640559105339892618506934230429, 11.51274542474059765268444625099, 12.60072765243956357571252404723

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