Properties

Label 2-177-59.4-c1-0-8
Degree $2$
Conductor $177$
Sign $0.989 + 0.142i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15 + 0.234i)2-s + (0.647 − 0.762i)3-s + (2.62 + 0.578i)4-s + (−1.34 − 1.02i)5-s + (1.57 − 1.48i)6-s + (−0.734 + 1.84i)7-s + (1.41 + 0.477i)8-s + (−0.161 − 0.986i)9-s + (−2.65 − 2.51i)10-s + (0.997 − 0.461i)11-s + (2.14 − 1.62i)12-s + (−0.907 + 5.53i)13-s + (−2.01 + 3.79i)14-s + (−1.65 + 0.363i)15-s + (−1.94 − 0.898i)16-s + (−0.674 − 1.69i)17-s + ⋯
L(s)  = 1  + (1.52 + 0.165i)2-s + (0.373 − 0.440i)3-s + (1.31 + 0.289i)4-s + (−0.602 − 0.457i)5-s + (0.641 − 0.608i)6-s + (−0.277 + 0.696i)7-s + (0.501 + 0.168i)8-s + (−0.0539 − 0.328i)9-s + (−0.840 − 0.796i)10-s + (0.300 − 0.139i)11-s + (0.618 − 0.470i)12-s + (−0.251 + 1.53i)13-s + (−0.537 + 1.01i)14-s + (−0.426 + 0.0938i)15-s + (−0.485 − 0.224i)16-s + (−0.163 − 0.410i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35501 - 0.168902i\)
\(L(\frac12)\) \(\approx\) \(2.35501 - 0.168902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.647 + 0.762i)T \)
59 \( 1 + (-4.82 + 5.97i)T \)
good2 \( 1 + (-2.15 - 0.234i)T + (1.95 + 0.429i)T^{2} \)
5 \( 1 + (1.34 + 1.02i)T + (1.33 + 4.81i)T^{2} \)
7 \( 1 + (0.734 - 1.84i)T + (-5.08 - 4.81i)T^{2} \)
11 \( 1 + (-0.997 + 0.461i)T + (7.12 - 8.38i)T^{2} \)
13 \( 1 + (0.907 - 5.53i)T + (-12.3 - 4.15i)T^{2} \)
17 \( 1 + (0.674 + 1.69i)T + (-12.3 + 11.6i)T^{2} \)
19 \( 1 + (1.30 + 1.92i)T + (-7.03 + 17.6i)T^{2} \)
23 \( 1 + (0.198 - 3.67i)T + (-22.8 - 2.48i)T^{2} \)
29 \( 1 + (-8.77 + 0.954i)T + (28.3 - 6.23i)T^{2} \)
31 \( 1 + (-0.663 + 0.977i)T + (-11.4 - 28.7i)T^{2} \)
37 \( 1 + (3.77 - 1.27i)T + (29.4 - 22.3i)T^{2} \)
41 \( 1 + (0.376 + 6.94i)T + (-40.7 + 4.43i)T^{2} \)
43 \( 1 + (-3.49 - 1.61i)T + (27.8 + 32.7i)T^{2} \)
47 \( 1 + (0.973 - 0.740i)T + (12.5 - 45.2i)T^{2} \)
53 \( 1 + (-3.09 + 2.93i)T + (2.86 - 52.9i)T^{2} \)
61 \( 1 + (13.9 + 1.51i)T + (59.5 + 13.1i)T^{2} \)
67 \( 1 + (-3.85 - 1.29i)T + (53.3 + 40.5i)T^{2} \)
71 \( 1 + (-11.1 + 8.50i)T + (18.9 - 68.4i)T^{2} \)
73 \( 1 + (5.01 - 9.46i)T + (-40.9 - 60.4i)T^{2} \)
79 \( 1 + (-9.88 - 11.6i)T + (-12.7 + 77.9i)T^{2} \)
83 \( 1 + (11.0 - 6.62i)T + (38.8 - 73.3i)T^{2} \)
89 \( 1 + (11.1 - 1.21i)T + (86.9 - 19.1i)T^{2} \)
97 \( 1 + (1.29 + 2.44i)T + (-54.4 + 80.2i)T^{2} \)
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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

−12.61522670408606820078625781139, −12.06574379220195976130913331076, −11.41256055516573288742998693968, −9.430230722284489436641544409044, −8.546627234517196125741465859228, −7.07682534913167249194267115646, −6.25298335711507416471777021920, −4.88425162836655733992697871758, −3.89691033095182424994106505746, −2.46375826516754364450389035841, 2.89346234832527682884258170022, 3.77011776165471627238588626005, 4.77193675702811119024508763380, 6.12036227888532839873795115081, 7.30128497882340832411471084288, 8.510802429346213599126596253924, 10.20538947571221880258634652773, 10.80619826972929965787161071903, 12.03269594171290471126056202767, 12.81102657886963597768074061570

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