Properties

Label 2-177-177.98-c1-0-14
Degree $2$
Conductor $177$
Sign $-0.507 + 0.861i$
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  + (−1.03 − 0.476i)2-s + (1.26 − 1.18i)3-s + (−0.460 − 0.542i)4-s + (−0.173 − 0.288i)5-s + (−1.86 + 0.612i)6-s + (−0.0891 − 1.64i)7-s + (0.823 + 2.96i)8-s + (0.210 − 2.99i)9-s + (0.0413 + 0.380i)10-s + (−0.710 − 4.33i)11-s + (−1.22 − 0.143i)12-s + (−3.33 + 4.38i)13-s + (−0.691 + 1.73i)14-s + (−0.561 − 0.160i)15-s + (0.334 − 2.04i)16-s + (4.91 + 0.266i)17-s + ⋯
L(s)  = 1  + (−0.728 − 0.336i)2-s + (0.731 − 0.681i)3-s + (−0.230 − 0.271i)4-s + (−0.0777 − 0.129i)5-s + (−0.762 + 0.250i)6-s + (−0.0337 − 0.621i)7-s + (0.291 + 1.04i)8-s + (0.0703 − 0.997i)9-s + (0.0130 + 0.120i)10-s + (−0.214 − 1.30i)11-s + (−0.353 − 0.0413i)12-s + (−0.924 + 1.21i)13-s + (−0.184 + 0.464i)14-s + (−0.144 − 0.0415i)15-s + (0.0837 − 0.510i)16-s + (1.19 + 0.0646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424788 - 0.742818i\)
\(L(\frac12)\) \(\approx\) \(0.424788 - 0.742818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.26 + 1.18i)T \)
59 \( 1 + (6.99 - 3.18i)T \)
good2 \( 1 + (1.03 + 0.476i)T + (1.29 + 1.52i)T^{2} \)
5 \( 1 + (0.173 + 0.288i)T + (-2.34 + 4.41i)T^{2} \)
7 \( 1 + (0.0891 + 1.64i)T + (-6.95 + 0.756i)T^{2} \)
11 \( 1 + (0.710 + 4.33i)T + (-10.4 + 3.51i)T^{2} \)
13 \( 1 + (3.33 - 4.38i)T + (-3.47 - 12.5i)T^{2} \)
17 \( 1 + (-4.91 - 0.266i)T + (16.9 + 1.83i)T^{2} \)
19 \( 1 + (3.01 + 2.85i)T + (1.02 + 18.9i)T^{2} \)
23 \( 1 + (-0.795 - 0.175i)T + (20.8 + 9.65i)T^{2} \)
29 \( 1 + (-0.998 - 2.15i)T + (-18.7 + 22.1i)T^{2} \)
31 \( 1 + (-2.90 - 3.07i)T + (-1.67 + 30.9i)T^{2} \)
37 \( 1 + (-6.35 - 1.76i)T + (31.7 + 19.0i)T^{2} \)
41 \( 1 + (-2.11 - 9.61i)T + (-37.2 + 17.2i)T^{2} \)
43 \( 1 + (-3.82 - 0.626i)T + (40.7 + 13.7i)T^{2} \)
47 \( 1 + (-7.13 - 4.29i)T + (22.0 + 41.5i)T^{2} \)
53 \( 1 + (-0.763 + 7.01i)T + (-51.7 - 11.3i)T^{2} \)
61 \( 1 + (3.49 - 7.54i)T + (-39.4 - 46.4i)T^{2} \)
67 \( 1 + (-1.07 + 0.297i)T + (57.4 - 34.5i)T^{2} \)
71 \( 1 + (-5.04 + 8.38i)T + (-33.2 - 62.7i)T^{2} \)
73 \( 1 + (7.66 + 3.05i)T + (52.9 + 50.2i)T^{2} \)
79 \( 1 + (-3.48 - 1.17i)T + (62.8 + 47.8i)T^{2} \)
83 \( 1 + (5.28 + 7.79i)T + (-30.7 + 77.1i)T^{2} \)
89 \( 1 + (11.1 - 5.13i)T + (57.6 - 67.8i)T^{2} \)
97 \( 1 + (-10.5 + 4.19i)T + (70.4 - 66.7i)T^{2} \)
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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

−12.31931653260513460774763261786, −11.26824840132235666680313255082, −10.20335041715523469108927744149, −9.220206404313069445581114843184, −8.440405165693225099890400717628, −7.49728189128991141887644326977, −6.20584366752018624107970999985, −4.51059542119288959637898233532, −2.74179919370253026023600378375, −0.999316244399477898429292903062, 2.66707886457367901313232121695, 4.11725261110653530819893653107, 5.41983241221980782769351893687, 7.41657191131676504341989758252, 7.900176568509607867238879723188, 9.051476006399301947320428980751, 9.852850109211008327156220867499, 10.47347345296638699691946648741, 12.32779482460029051750053326402, 12.82611352763336436706397015793

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