Properties

Label 2-177-177.5-c2-0-28
Degree $2$
Conductor $177$
Sign $0.632 + 0.774i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.892 + 2.64i)2-s + (0.713 − 2.91i)3-s + (−3.03 − 2.30i)4-s + (−2.87 − 1.14i)5-s + (7.08 + 4.48i)6-s + (2.33 − 1.08i)7-s + (−0.437 + 0.296i)8-s + (−7.98 − 4.15i)9-s + (5.59 − 6.58i)10-s + (−5.90 − 1.64i)11-s + (−8.88 + 7.19i)12-s + (11.2 − 21.3i)13-s + (0.778 + 7.15i)14-s + (−5.38 + 7.55i)15-s + (−4.47 − 16.1i)16-s + (8.75 − 18.9i)17-s + ⋯
L(s)  = 1  + (−0.446 + 1.32i)2-s + (0.237 − 0.971i)3-s + (−0.758 − 0.576i)4-s + (−0.574 − 0.228i)5-s + (1.18 + 0.748i)6-s + (0.333 − 0.154i)7-s + (−0.0547 + 0.0371i)8-s + (−0.886 − 0.461i)9-s + (0.559 − 0.658i)10-s + (−0.537 − 0.149i)11-s + (−0.740 + 0.599i)12-s + (0.869 − 1.63i)13-s + (0.0555 + 0.511i)14-s + (−0.358 + 0.503i)15-s + (−0.279 − 1.00i)16-s + (0.514 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.632 + 0.774i$
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: \((2,\ 177,\ (\ :1),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.771389 - 0.365732i\)
\(L(\frac12)\) \(\approx\) \(0.771389 - 0.365732i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.713 + 2.91i)T \)
59 \( 1 + (5.88 + 58.7i)T \)
good2 \( 1 + (0.892 - 2.64i)T + (-3.18 - 2.42i)T^{2} \)
5 \( 1 + (2.87 + 1.14i)T + (18.1 + 17.1i)T^{2} \)
7 \( 1 + (-2.33 + 1.08i)T + (31.7 - 37.3i)T^{2} \)
11 \( 1 + (5.90 + 1.64i)T + (103. + 62.3i)T^{2} \)
13 \( 1 + (-11.2 + 21.3i)T + (-94.8 - 139. i)T^{2} \)
17 \( 1 + (-8.75 + 18.9i)T + (-187. - 220. i)T^{2} \)
19 \( 1 + (-1.82 + 0.402i)T + (327. - 151. i)T^{2} \)
23 \( 1 + (21.5 + 3.52i)T + (501. + 168. i)T^{2} \)
29 \( 1 + (-2.19 - 6.51i)T + (-669. + 508. i)T^{2} \)
31 \( 1 + (-44.5 - 9.81i)T + (872. + 403. i)T^{2} \)
37 \( 1 + (32.5 - 47.9i)T + (-506. - 1.27e3i)T^{2} \)
41 \( 1 + (-4.66 + 0.765i)T + (1.59e3 - 536. i)T^{2} \)
43 \( 1 + (12.9 + 46.5i)T + (-1.58e3 + 953. i)T^{2} \)
47 \( 1 + (-22.9 + 9.15i)T + (1.60e3 - 1.51e3i)T^{2} \)
53 \( 1 + (4.66 - 3.96i)T + (454. - 2.77e3i)T^{2} \)
61 \( 1 + (-3.40 - 1.14i)T + (2.96e3 + 2.25e3i)T^{2} \)
67 \( 1 + (-27.0 - 39.9i)T + (-1.66e3 + 4.17e3i)T^{2} \)
71 \( 1 + (32.5 - 12.9i)T + (3.65e3 - 3.46e3i)T^{2} \)
73 \( 1 + (-20.7 + 2.25i)T + (5.20e3 - 1.14e3i)T^{2} \)
79 \( 1 + (82.7 - 49.7i)T + (2.92e3 - 5.51e3i)T^{2} \)
83 \( 1 + (-104. + 5.66i)T + (6.84e3 - 744. i)T^{2} \)
89 \( 1 + (-6.55 - 19.4i)T + (-6.30e3 + 4.79e3i)T^{2} \)
97 \( 1 + (83.6 + 9.09i)T + (9.18e3 + 2.02e3i)T^{2} \)
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

−12.37465085817041439061992680726, −11.55600097776365091250536165670, −10.10030399374119927888884593796, −8.495434393645654064384158654550, −8.124823989606157384255101705674, −7.36718395292165937943979958335, −6.19801151300222428408638986733, −5.20941899552980508480497604061, −3.05062108283305763339708461325, −0.57533168057218859014601428482, 1.97955214246138884484815084204, 3.51438181474313508731999345223, 4.31445594802235958264084365306, 6.09201000757209297133399995610, 7.988472026352566660728791573933, 8.874039492553844448591792658748, 9.834036455323362060852107648684, 10.66041878743297028194764841258, 11.45015740970696410626435393689, 12.04135585375474235134557101688

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