Properties

Label 2-177-59.58-c2-0-17
Degree $2$
Conductor $177$
Sign $-0.892 + 0.451i$
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  − 1.39i·2-s − 1.73·3-s + 2.04·4-s + 0.273·5-s + 2.42i·6-s − 10.8·7-s − 8.45i·8-s + 2.99·9-s − 0.382i·10-s − 15.0i·11-s − 3.53·12-s − 11.1i·13-s + 15.2i·14-s − 0.473·15-s − 3.67·16-s − 3.40·17-s + ⋯
L(s)  = 1  − 0.699i·2-s − 0.577·3-s + 0.510·4-s + 0.0546·5-s + 0.404i·6-s − 1.55·7-s − 1.05i·8-s + 0.333·9-s − 0.0382i·10-s − 1.36i·11-s − 0.294·12-s − 0.854i·13-s + 1.08i·14-s − 0.0315·15-s − 0.229·16-s − 0.200·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.212145 - 0.889357i\)
\(L(\frac12)\) \(\approx\) \(0.212145 - 0.889357i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
59 \( 1 + (-52.6 + 26.6i)T \)
good2 \( 1 + 1.39iT - 4T^{2} \)
5 \( 1 - 0.273T + 25T^{2} \)
7 \( 1 + 10.8T + 49T^{2} \)
11 \( 1 + 15.0iT - 121T^{2} \)
13 \( 1 + 11.1iT - 169T^{2} \)
17 \( 1 + 3.40T + 289T^{2} \)
19 \( 1 + 15.7T + 361T^{2} \)
23 \( 1 - 15.2iT - 529T^{2} \)
29 \( 1 - 22.5T + 841T^{2} \)
31 \( 1 + 59.5iT - 961T^{2} \)
37 \( 1 - 42.4iT - 1.36e3T^{2} \)
41 \( 1 - 78.4T + 1.68e3T^{2} \)
43 \( 1 - 78.3iT - 1.84e3T^{2} \)
47 \( 1 + 35.3iT - 2.20e3T^{2} \)
53 \( 1 - 37.2T + 2.80e3T^{2} \)
61 \( 1 + 44.2iT - 3.72e3T^{2} \)
67 \( 1 - 17.3iT - 4.48e3T^{2} \)
71 \( 1 + 59.0T + 5.04e3T^{2} \)
73 \( 1 + 85.3iT - 5.32e3T^{2} \)
79 \( 1 - 62.2T + 6.24e3T^{2} \)
83 \( 1 + 98.9iT - 6.88e3T^{2} \)
89 \( 1 - 35.9iT - 7.92e3T^{2} \)
97 \( 1 + 11.0iT - 9.40e3T^{2} \)
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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

−11.89794614160565497883313072138, −11.10511316413289622243196151001, −10.21721881388114171792829577040, −9.443934788317760944323590775911, −7.84513087412966745116953434338, −6.38146116138623002885814171602, −5.95352633347708786964406351127, −3.80921165818730600022347011305, −2.74405032590947874509790643102, −0.55100635932318127334360703787, 2.32402830146647060804930381053, 4.25123671688110568579732737773, 5.75657610216674086005574869841, 6.73096032392147728084805305949, 7.16614224725934258955610444292, 8.835741487013820617554500818646, 9.950851019208396845051464985360, 10.80464523404307579405562453891, 12.17845632408353368225434463814, 12.58257779690237574070673029862

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