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Properties

Label 2-177-3.2-c2-0-6
Degree $2$
Conductor $177$
Sign $0.781 - 0.623i$
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  − 3.09i·2-s + (1.87 + 2.34i)3-s − 5.59·4-s + 8.72i·5-s + (7.26 − 5.79i)6-s − 11.2·7-s + 4.93i·8-s + (−1.99 + 8.77i)9-s + 27.0·10-s + 9.18i·11-s + (−10.4 − 13.1i)12-s + 7.17·13-s + 34.7i·14-s + (−20.4 + 16.3i)15-s − 7.08·16-s + 10.0i·17-s + ⋯
L(s)  = 1  − 1.54i·2-s + (0.623 + 0.781i)3-s − 1.39·4-s + 1.74i·5-s + (1.21 − 0.965i)6-s − 1.60·7-s + 0.617i·8-s + (−0.221 + 0.975i)9-s + 2.70·10-s + 0.834i·11-s + (−0.872 − 1.09i)12-s + 0.551·13-s + 2.48i·14-s + (−1.36 + 1.08i)15-s − 0.442·16-s + 0.590i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.781 - 0.623i$
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: \((2,\ 177,\ (\ :1),\ 0.781 - 0.623i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20244 + 0.420953i\)
\(L(\frac12)\) \(\approx\) \(1.20244 + 0.420953i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.87 - 2.34i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 3.09iT - 4T^{2} \)
5 \( 1 - 8.72iT - 25T^{2} \)
7 \( 1 + 11.2T + 49T^{2} \)
11 \( 1 - 9.18iT - 121T^{2} \)
13 \( 1 - 7.17T + 169T^{2} \)
17 \( 1 - 10.0iT - 289T^{2} \)
19 \( 1 - 27.7T + 361T^{2} \)
23 \( 1 + 2.46iT - 529T^{2} \)
29 \( 1 + 42.9iT - 841T^{2} \)
31 \( 1 - 39.9T + 961T^{2} \)
37 \( 1 + 42.6T + 1.36e3T^{2} \)
41 \( 1 + 20.2iT - 1.68e3T^{2} \)
43 \( 1 - 56.9T + 1.84e3T^{2} \)
47 \( 1 - 47.6iT - 2.20e3T^{2} \)
53 \( 1 - 68.4iT - 2.80e3T^{2} \)
61 \( 1 + 31.4T + 3.72e3T^{2} \)
67 \( 1 - 17.2T + 4.48e3T^{2} \)
71 \( 1 - 42.1iT - 5.04e3T^{2} \)
73 \( 1 - 45.5T + 5.32e3T^{2} \)
79 \( 1 - 27.8T + 6.24e3T^{2} \)
83 \( 1 - 0.101iT - 6.88e3T^{2} \)
89 \( 1 + 34.2iT - 7.92e3T^{2} \)
97 \( 1 - 59.1T + 9.40e3T^{2} \)
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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

−12.38953028017564413985828035543, −11.32326989486838329183577054427, −10.34946241148956147878123534963, −10.00293088697666930206580546063, −9.235943881263900439593934466265, −7.44767764391800677306607560148, −6.21772487313621301452087682106, −4.03462259862840858005966210893, −3.21367901097278985946324725972, −2.50447313377578486605655206130, 0.72061963044604744534574327416, 3.42886288565718987061009368150, 5.20415374963244135315048335731, 6.13818123739570306186366681126, 7.11875359009413283681192511102, 8.240797392926697218014307185131, 8.943681846543709975885650879814, 9.551723110967925492912832234167, 11.88403346047008336074738409735, 12.79766493523062159632304157500

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