Properties

Label 2-177-3.2-c2-0-37
Degree $2$
Conductor $177$
Sign $0.240 - 0.970i$
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  − 3.49i·2-s + (−2.91 − 0.722i)3-s − 8.23·4-s − 8.21i·5-s + (−2.52 + 10.1i)6-s − 3.56·7-s + 14.7i·8-s + (7.95 + 4.20i)9-s − 28.7·10-s − 6.15i·11-s + (23.9 + 5.95i)12-s + 2.36·13-s + 12.4i·14-s + (−5.93 + 23.9i)15-s + 18.8·16-s + 0.880i·17-s + ⋯
L(s)  = 1  − 1.74i·2-s + (−0.970 − 0.240i)3-s − 2.05·4-s − 1.64i·5-s + (−0.421 + 1.69i)6-s − 0.509·7-s + 1.84i·8-s + (0.883 + 0.467i)9-s − 2.87·10-s − 0.559i·11-s + (1.99 + 0.495i)12-s + 0.181·13-s + 0.890i·14-s + (−0.395 + 1.59i)15-s + 1.17·16-s + 0.0518i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.240 - 0.970i$
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: \((2,\ 177,\ (\ :1),\ 0.240 - 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.478982 + 0.374606i\)
\(L(\frac12)\) \(\approx\) \(0.478982 + 0.374606i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.91 + 0.722i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 + 3.49iT - 4T^{2} \)
5 \( 1 + 8.21iT - 25T^{2} \)
7 \( 1 + 3.56T + 49T^{2} \)
11 \( 1 + 6.15iT - 121T^{2} \)
13 \( 1 - 2.36T + 169T^{2} \)
17 \( 1 - 0.880iT - 289T^{2} \)
19 \( 1 - 32.8T + 361T^{2} \)
23 \( 1 - 20.2iT - 529T^{2} \)
29 \( 1 + 46.3iT - 841T^{2} \)
31 \( 1 + 55.1T + 961T^{2} \)
37 \( 1 + 10.4T + 1.36e3T^{2} \)
41 \( 1 + 5.03iT - 1.68e3T^{2} \)
43 \( 1 - 14.2T + 1.84e3T^{2} \)
47 \( 1 - 75.4iT - 2.20e3T^{2} \)
53 \( 1 + 72.2iT - 2.80e3T^{2} \)
61 \( 1 + 8.24T + 3.72e3T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 + 97.6iT - 5.04e3T^{2} \)
73 \( 1 - 28.7T + 5.32e3T^{2} \)
79 \( 1 - 149.T + 6.24e3T^{2} \)
83 \( 1 - 85.1iT - 6.88e3T^{2} \)
89 \( 1 + 98.6iT - 7.92e3T^{2} \)
97 \( 1 - 93.2T + 9.40e3T^{2} \)
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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

−11.76415890871543302439660805884, −11.02761871491358221563349474118, −9.709101191972140828455232733467, −9.244646117041255249944824496184, −7.82075460055507341779947272901, −5.76042072073322784078482796688, −4.87181457668711798283559319118, −3.63518592935291555292408347374, −1.53887462434513867732719599595, −0.44304133999578444320141032758, 3.55789839340438000000454372635, 5.12693346303842338032183267166, 6.09787870165858339816587896582, 7.01851990905733442740014271044, 7.36708151088596333593308623596, 9.188285157053197117141176177399, 10.14957353496201003530650151092, 11.07159098445521735761427516743, 12.37278219546345169857118333098, 13.60713689688643928547518072422

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