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Properties

Label 2-177-3.2-c2-0-17
Degree $2$
Conductor $177$
Sign $0.110 + 0.993i$
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  − 2.12i·2-s + (−2.98 + 0.332i)3-s − 0.528·4-s + 5.15i·5-s + (0.707 + 6.34i)6-s + 2.69·7-s − 7.38i·8-s + (8.77 − 1.98i)9-s + 10.9·10-s − 15.4i·11-s + (1.57 − 0.175i)12-s + 18.6·13-s − 5.72i·14-s + (−1.71 − 15.3i)15-s − 17.8·16-s + 4.88i·17-s + ⋯
L(s)  = 1  − 1.06i·2-s + (−0.993 + 0.110i)3-s − 0.132·4-s + 1.03i·5-s + (0.117 + 1.05i)6-s + 0.384·7-s − 0.923i·8-s + (0.975 − 0.220i)9-s + 1.09·10-s − 1.40i·11-s + (0.131 − 0.0146i)12-s + 1.43·13-s − 0.409i·14-s + (−0.114 − 1.02i)15-s − 1.11·16-s + 0.287i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.110 + 0.993i$
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: \((2,\ 177,\ (\ :1),\ 0.110 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.966943 - 0.865053i\)
\(L(\frac12)\) \(\approx\) \(0.966943 - 0.865053i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.98 - 0.332i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 2.12iT - 4T^{2} \)
5 \( 1 - 5.15iT - 25T^{2} \)
7 \( 1 - 2.69T + 49T^{2} \)
11 \( 1 + 15.4iT - 121T^{2} \)
13 \( 1 - 18.6T + 169T^{2} \)
17 \( 1 - 4.88iT - 289T^{2} \)
19 \( 1 - 3.41T + 361T^{2} \)
23 \( 1 + 15.9iT - 529T^{2} \)
29 \( 1 + 24.8iT - 841T^{2} \)
31 \( 1 - 28.0T + 961T^{2} \)
37 \( 1 - 60.2T + 1.36e3T^{2} \)
41 \( 1 - 37.6iT - 1.68e3T^{2} \)
43 \( 1 + 15.8T + 1.84e3T^{2} \)
47 \( 1 + 13.1iT - 2.20e3T^{2} \)
53 \( 1 - 35.0iT - 2.80e3T^{2} \)
61 \( 1 + 83.7T + 3.72e3T^{2} \)
67 \( 1 - 72.8T + 4.48e3T^{2} \)
71 \( 1 - 27.7iT - 5.04e3T^{2} \)
73 \( 1 - 54.4T + 5.32e3T^{2} \)
79 \( 1 + 123.T + 6.24e3T^{2} \)
83 \( 1 - 143. iT - 6.88e3T^{2} \)
89 \( 1 + 21.8iT - 7.92e3T^{2} \)
97 \( 1 + 98.9T + 9.40e3T^{2} \)
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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

−11.76096659689974167702293032995, −11.06232722815185548559673965719, −10.83117989363319274178805199336, −9.753629432463632265057837857571, −8.186194639712562349187513283527, −6.60380316749491153935177332743, −6.01464591315058896903286780501, −4.13841856234141466642267226766, −2.95745887874184649452005842311, −1.05765986099746576979933698929, 1.48625283974917872200328538798, 4.50574855630646126701970525128, 5.28975411688002147462757139372, 6.33811035358476882402620649273, 7.35240087181283330436015579874, 8.330073281218276815081411989297, 9.525776775863805581641718098278, 10.89946691764108517844063542112, 11.74918604676787927215440354284, 12.67400951976303250703823591907

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