Properties

Label 2-177-3.2-c4-0-50
Degree $2$
Conductor $177$
Sign $-0.903 + 0.428i$
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  − 5.04i·2-s + (−3.85 − 8.13i)3-s − 9.49·4-s + 33.9i·5-s + (−41.0 + 19.4i)6-s + 88.4·7-s − 32.8i·8-s + (−51.2 + 62.6i)9-s + 171.·10-s − 111. i·11-s + (36.5 + 77.2i)12-s + 6.16·13-s − 446. i·14-s + (276. − 130. i)15-s − 317.·16-s − 288. i·17-s + ⋯
L(s)  = 1  − 1.26i·2-s + (−0.428 − 0.903i)3-s − 0.593·4-s + 1.35i·5-s + (−1.14 + 0.540i)6-s + 1.80·7-s − 0.513i·8-s + (−0.633 + 0.773i)9-s + 1.71·10-s − 0.920i·11-s + (0.254 + 0.536i)12-s + 0.0365·13-s − 2.27i·14-s + (1.22 − 0.581i)15-s − 1.24·16-s − 0.998i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.903 + 0.428i$
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: \((2,\ 177,\ (\ :2),\ -0.903 + 0.428i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.904947094\)
\(L(\frac12)\) \(\approx\) \(1.904947094\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.85 + 8.13i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 5.04iT - 16T^{2} \)
5 \( 1 - 33.9iT - 625T^{2} \)
7 \( 1 - 88.4T + 2.40e3T^{2} \)
11 \( 1 + 111. iT - 1.46e4T^{2} \)
13 \( 1 - 6.16T + 2.85e4T^{2} \)
17 \( 1 + 288. iT - 8.35e4T^{2} \)
19 \( 1 - 261.T + 1.30e5T^{2} \)
23 \( 1 + 401. iT - 2.79e5T^{2} \)
29 \( 1 + 696. iT - 7.07e5T^{2} \)
31 \( 1 - 1.23e3T + 9.23e5T^{2} \)
37 \( 1 + 1.13e3T + 1.87e6T^{2} \)
41 \( 1 - 1.50e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.23e3T + 3.41e6T^{2} \)
47 \( 1 - 148. iT - 4.87e6T^{2} \)
53 \( 1 + 4.97e3iT - 7.89e6T^{2} \)
61 \( 1 - 2.56e3T + 1.38e7T^{2} \)
67 \( 1 + 7.69e3T + 2.01e7T^{2} \)
71 \( 1 + 7.59e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.75e3T + 2.83e7T^{2} \)
79 \( 1 - 8.16e3T + 3.89e7T^{2} \)
83 \( 1 - 7.86e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.35e3iT - 6.27e7T^{2} \)
97 \( 1 + 7.98e3T + 8.85e7T^{2} \)
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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

−11.46507661031339227768097830249, −11.05224278446232936885885233299, −10.15475647099070103799225739740, −8.419017640332863003808135667784, −7.41573572823231721835021969135, −6.38685976622754012496642311316, −4.88635463423424614113380259008, −3.08097897037988451949769243172, −2.10254453731637463207067738406, −0.817555016764292438359204302697, 1.48946759732154978864191690757, 4.39710441962844760674629491481, 4.99767161667536451379395451856, 5.72422841115472304133196973133, 7.37684153303782581894194397592, 8.391653886153340699639102994119, 8.974307877715517144728876568738, 10.38440742221028365450407277481, 11.54692496953469000730261376807, 12.23839740977942609688217399342

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