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Properties

Label 2-177-3.2-c4-0-0
Degree $2$
Conductor $177$
Sign $0.630 - 0.775i$
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  − 4.27i·2-s + (−6.98 − 5.67i)3-s − 2.31·4-s − 6.15i·5-s + (−24.2 + 29.8i)6-s − 22.1·7-s − 58.5i·8-s + (16.5 + 79.2i)9-s − 26.3·10-s − 60.0i·11-s + (16.1 + 13.1i)12-s − 252.·13-s + 94.7i·14-s + (−34.9 + 43.0i)15-s − 287.·16-s + 475. i·17-s + ⋯
L(s)  = 1  − 1.06i·2-s + (−0.775 − 0.630i)3-s − 0.144·4-s − 0.246i·5-s + (−0.674 + 0.830i)6-s − 0.452·7-s − 0.915i·8-s + (0.204 + 0.978i)9-s − 0.263·10-s − 0.496i·11-s + (0.112 + 0.0911i)12-s − 1.49·13-s + 0.483i·14-s + (−0.155 + 0.191i)15-s − 1.12·16-s + 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.630 - 0.775i$
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: \((2,\ 177,\ (\ :2),\ 0.630 - 0.775i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.06501468774\)
\(L(\frac12)\) \(\approx\) \(0.06501468774\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.98 + 5.67i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 4.27iT - 16T^{2} \)
5 \( 1 + 6.15iT - 625T^{2} \)
7 \( 1 + 22.1T + 2.40e3T^{2} \)
11 \( 1 + 60.0iT - 1.46e4T^{2} \)
13 \( 1 + 252.T + 2.85e4T^{2} \)
17 \( 1 - 475. iT - 8.35e4T^{2} \)
19 \( 1 + 114.T + 1.30e5T^{2} \)
23 \( 1 - 529. iT - 2.79e5T^{2} \)
29 \( 1 + 1.25e3iT - 7.07e5T^{2} \)
31 \( 1 - 14.1T + 9.23e5T^{2} \)
37 \( 1 + 682.T + 1.87e6T^{2} \)
41 \( 1 - 1.78e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.89e3T + 3.41e6T^{2} \)
47 \( 1 - 742. iT - 4.87e6T^{2} \)
53 \( 1 - 3.07e3iT - 7.89e6T^{2} \)
61 \( 1 - 1.43e3T + 1.38e7T^{2} \)
67 \( 1 - 3.22e3T + 2.01e7T^{2} \)
71 \( 1 + 5.48e3iT - 2.54e7T^{2} \)
73 \( 1 + 8.85e3T + 2.83e7T^{2} \)
79 \( 1 + 9.34e3T + 3.89e7T^{2} \)
83 \( 1 - 7.55e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.86e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.25e4T + 8.85e7T^{2} \)
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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

−12.20723357905635878527304545464, −11.25672790292899217456165313482, −10.43487306024856498630254756393, −9.550565441644086224445945265734, −7.982072243514631975189735000498, −6.82035732913508295423501217282, −5.80353555870100428081827104079, −4.29104376013222680944989336806, −2.68753347557582489881545022327, −1.39704509735030503646375768133, 0.02618018573224843408213491244, 2.73238073267605951250725054229, 4.69012868080369582988565546619, 5.40395067061750420077463946840, 6.86567596852032890907365213651, 7.10899921203520180475552644708, 8.836515581195838693098400781829, 9.849386944883628950035541075400, 10.80746553688410912613941868244, 11.87804894537002398809208899768

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