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Properties

Label 2-177-3.2-c4-0-17
Degree $2$
Conductor $177$
Sign $-0.608 + 0.793i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.62i·2-s + (7.14 + 5.47i)3-s − 5.42·4-s + 33.7i·5-s + (−25.3 + 33.0i)6-s − 90.1·7-s + 48.9i·8-s + (21.0 + 78.2i)9-s − 156.·10-s − 130. i·11-s + (−38.7 − 29.7i)12-s + 134.·13-s − 417. i·14-s + (−184. + 240. i)15-s − 313.·16-s − 178. i·17-s + ⋯
L(s)  = 1  + 1.15i·2-s + (0.793 + 0.608i)3-s − 0.339·4-s + 1.34i·5-s + (−0.704 + 0.918i)6-s − 1.84·7-s + 0.764i·8-s + (0.259 + 0.965i)9-s − 1.56·10-s − 1.07i·11-s + (−0.269 − 0.206i)12-s + 0.797·13-s − 2.12i·14-s + (−0.820 + 1.06i)15-s − 1.22·16-s − 0.616i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.608 + 0.793i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.758136581\)
\(L(\frac12)\) \(\approx\) \(1.758136581\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.14 - 5.47i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 - 4.62iT - 16T^{2} \)
5 \( 1 - 33.7iT - 625T^{2} \)
7 \( 1 + 90.1T + 2.40e3T^{2} \)
11 \( 1 + 130. iT - 1.46e4T^{2} \)
13 \( 1 - 134.T + 2.85e4T^{2} \)
17 \( 1 + 178. iT - 8.35e4T^{2} \)
19 \( 1 - 400.T + 1.30e5T^{2} \)
23 \( 1 + 68.8iT - 2.79e5T^{2} \)
29 \( 1 - 1.46e3iT - 7.07e5T^{2} \)
31 \( 1 + 240.T + 9.23e5T^{2} \)
37 \( 1 - 223.T + 1.87e6T^{2} \)
41 \( 1 + 200. iT - 2.82e6T^{2} \)
43 \( 1 + 3.01e3T + 3.41e6T^{2} \)
47 \( 1 - 2.56e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.07e3iT - 7.89e6T^{2} \)
61 \( 1 - 3.52e3T + 1.38e7T^{2} \)
67 \( 1 + 5.17e3T + 2.01e7T^{2} \)
71 \( 1 - 3.02e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.15e3T + 2.83e7T^{2} \)
79 \( 1 + 1.12e4T + 3.89e7T^{2} \)
83 \( 1 - 9.46e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.70e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.34e3T + 8.85e7T^{2} \)
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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

−13.19237086084286207765540425457, −11.36730698397617205641753733648, −10.49286552073458825821485813099, −9.474982094029939188906684652765, −8.515714461481024074279309166935, −7.22111447248643436346610430238, −6.59000526001344358744367373092, −5.54385811765791052206679967699, −3.40253816022710662387294747377, −2.94227234826622210187910077895, 0.57231221824026932117631861843, 1.77012661201785470079198178474, 3.15773316933078394447447586387, 4.11583428347435059726411965315, 6.12527828963605732036182396559, 7.23673758023181289310273985365, 8.661341157867530019196190374432, 9.594271379090698995378047100062, 9.993808627148156874066335595045, 11.84367882510080668818872773680

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