Properties

Label 2-177-177.5-c2-0-3
Degree $2$
Conductor $177$
Sign $-0.612 + 0.790i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.912 + 2.70i)2-s + (2.96 − 0.473i)3-s + (−3.31 − 2.51i)4-s + (−6.32 − 2.51i)5-s + (−1.42 + 8.45i)6-s + (−6.08 + 2.81i)7-s + (0.377 − 0.255i)8-s + (8.55 − 2.80i)9-s + (12.5 − 14.8i)10-s + (−16.9 − 4.70i)11-s + (−11.0 − 5.88i)12-s + (−4.39 + 8.28i)13-s + (−2.07 − 19.0i)14-s + (−19.9 − 4.46i)15-s + (−4.10 − 14.7i)16-s + (−11.2 + 24.3i)17-s + ⋯
L(s)  = 1  + (−0.456 + 1.35i)2-s + (0.987 − 0.157i)3-s + (−0.827 − 0.629i)4-s + (−1.26 − 0.503i)5-s + (−0.236 + 1.40i)6-s + (−0.869 + 0.402i)7-s + (0.0471 − 0.0319i)8-s + (0.950 − 0.311i)9-s + (1.25 − 1.48i)10-s + (−1.54 − 0.427i)11-s + (−0.916 − 0.490i)12-s + (−0.337 + 0.637i)13-s + (−0.147 − 1.36i)14-s + (−1.32 − 0.297i)15-s + (−0.256 − 0.923i)16-s + (−0.663 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.134922 - 0.275283i\)
\(L(\frac12)\) \(\approx\) \(0.134922 - 0.275283i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.96 + 0.473i)T \)
59 \( 1 + (-4.27 + 58.8i)T \)
good2 \( 1 + (0.912 - 2.70i)T + (-3.18 - 2.42i)T^{2} \)
5 \( 1 + (6.32 + 2.51i)T + (18.1 + 17.1i)T^{2} \)
7 \( 1 + (6.08 - 2.81i)T + (31.7 - 37.3i)T^{2} \)
11 \( 1 + (16.9 + 4.70i)T + (103. + 62.3i)T^{2} \)
13 \( 1 + (4.39 - 8.28i)T + (-94.8 - 139. i)T^{2} \)
17 \( 1 + (11.2 - 24.3i)T + (-187. - 220. i)T^{2} \)
19 \( 1 + (-2.20 + 0.485i)T + (327. - 151. i)T^{2} \)
23 \( 1 + (-20.8 - 3.42i)T + (501. + 168. i)T^{2} \)
29 \( 1 + (-1.66 - 4.93i)T + (-669. + 508. i)T^{2} \)
31 \( 1 + (20.0 + 4.41i)T + (872. + 403. i)T^{2} \)
37 \( 1 + (-19.4 + 28.6i)T + (-506. - 1.27e3i)T^{2} \)
41 \( 1 + (-63.9 + 10.4i)T + (1.59e3 - 536. i)T^{2} \)
43 \( 1 + (-8.90 - 32.0i)T + (-1.58e3 + 953. i)T^{2} \)
47 \( 1 + (24.4 - 9.72i)T + (1.60e3 - 1.51e3i)T^{2} \)
53 \( 1 + (73.6 - 62.5i)T + (454. - 2.77e3i)T^{2} \)
61 \( 1 + (28.7 + 9.69i)T + (2.96e3 + 2.25e3i)T^{2} \)
67 \( 1 + (0.0239 + 0.0353i)T + (-1.66e3 + 4.17e3i)T^{2} \)
71 \( 1 + (72.5 - 28.8i)T + (3.65e3 - 3.46e3i)T^{2} \)
73 \( 1 + (111. - 12.1i)T + (5.20e3 - 1.14e3i)T^{2} \)
79 \( 1 + (89.3 - 53.7i)T + (2.92e3 - 5.51e3i)T^{2} \)
83 \( 1 + (39.6 - 2.14i)T + (6.84e3 - 744. i)T^{2} \)
89 \( 1 + (-10.4 - 31.0i)T + (-6.30e3 + 4.79e3i)T^{2} \)
97 \( 1 + (-58.6 - 6.37i)T + (9.18e3 + 2.02e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.03372471845118013103545462589, −12.57600604897256392459378369890, −11.01971593273487280897787229268, −9.468019858426293619893452996798, −8.704060218345907081901627796171, −7.936899687311496475505421987139, −7.27391587408427094705981480345, −6.00609843034156559959315158161, −4.44172668796606650972655115219, −2.92196876057841099405237236049, 0.17810770028630353608146400785, 2.70803515479287709796129307410, 3.20826305807605012655903886889, 4.56221660547384694597359613512, 7.10915628297010117825256738347, 7.81533403628900908847481131496, 9.072324662187127989781800488861, 10.01547092911757616051354782265, 10.67944489823112661433587830959, 11.63766290625633838009007042774

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