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Properties

Label 2-177-177.176-c3-0-15
Degree $2$
Conductor $177$
Sign $0.744 - 0.667i$
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  − 4.28·2-s + (0.191 − 5.19i)3-s + 10.3·4-s + 4.91i·5-s + (−0.821 + 22.2i)6-s + 31.8·7-s − 10.1·8-s + (−26.9 − 1.99i)9-s − 21.0i·10-s − 38.0·11-s + (1.98 − 53.8i)12-s + 69.1i·13-s − 136.·14-s + (25.5 + 0.942i)15-s − 39.3·16-s + 38.2i·17-s + ⋯
L(s)  = 1  − 1.51·2-s + (0.0368 − 0.999i)3-s + 1.29·4-s + 0.439i·5-s + (−0.0559 + 1.51i)6-s + 1.71·7-s − 0.450·8-s + (−0.997 − 0.0737i)9-s − 0.666i·10-s − 1.04·11-s + (0.0478 − 1.29i)12-s + 1.47i·13-s − 2.60·14-s + (0.439 + 0.0162i)15-s − 0.614·16-s + 0.545i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.669741 + 0.256320i\)
\(L(\frac12)\) \(\approx\) \(0.669741 + 0.256320i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.191 + 5.19i)T \)
59 \( 1 + (-314. - 326. i)T \)
good2 \( 1 + 4.28T + 8T^{2} \)
5 \( 1 - 4.91iT - 125T^{2} \)
7 \( 1 - 31.8T + 343T^{2} \)
11 \( 1 + 38.0T + 1.33e3T^{2} \)
13 \( 1 - 69.1iT - 2.19e3T^{2} \)
17 \( 1 - 38.2iT - 4.91e3T^{2} \)
19 \( 1 + 79.5T + 6.85e3T^{2} \)
23 \( 1 - 61.2T + 1.21e4T^{2} \)
29 \( 1 - 168. iT - 2.43e4T^{2} \)
31 \( 1 - 22.7iT - 2.97e4T^{2} \)
37 \( 1 - 197. iT - 5.06e4T^{2} \)
41 \( 1 - 415. iT - 6.89e4T^{2} \)
43 \( 1 + 429. iT - 7.95e4T^{2} \)
47 \( 1 - 90.7T + 1.03e5T^{2} \)
53 \( 1 + 4.32iT - 1.48e5T^{2} \)
61 \( 1 - 230. iT - 2.26e5T^{2} \)
67 \( 1 + 803. iT - 3.00e5T^{2} \)
71 \( 1 + 606. iT - 3.57e5T^{2} \)
73 \( 1 - 723. iT - 3.89e5T^{2} \)
79 \( 1 + 282.T + 4.93e5T^{2} \)
83 \( 1 + 375.T + 5.71e5T^{2} \)
89 \( 1 - 1.44e3T + 7.04e5T^{2} \)
97 \( 1 - 1.44e3iT - 9.12e5T^{2} \)
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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

−11.95584487176587527398610076763, −11.06718801487226125572810009529, −10.56905096479991848705647034689, −8.886120348732049883711729382526, −8.333725587362031977302956996453, −7.45147204079503012835282202765, −6.62111999755813386259242179314, −4.86968958144465366018001915527, −2.29328986615241315429057346829, −1.36610693030237073832136528389, 0.57316405969928163283538866352, 2.42942656579408299334672431173, 4.59479547495355499577602609696, 5.46011365042699240177900280079, 7.62587752985981057773196444278, 8.261475951780206144032438519921, 8.938860480627422092868382691434, 10.19955510792440462334288321297, 10.76407418189552809961788395580, 11.46290774902067336434210429041

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