Properties

Label 2-177-59.22-c1-0-6
Degree $2$
Conductor $177$
Sign $-0.114 + 0.993i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.335 − 2.04i)2-s + (0.267 + 0.963i)3-s + (−2.17 + 0.733i)4-s + (0.939 − 1.38i)5-s + (1.88 − 0.870i)6-s + (4.70 + 1.03i)7-s + (0.288 + 0.543i)8-s + (−0.856 + 0.515i)9-s + (−3.15 − 1.45i)10-s + (−1.41 − 1.07i)11-s + (−1.28 − 1.90i)12-s + (−3.26 − 1.96i)13-s + (0.540 − 9.96i)14-s + (1.58 + 0.534i)15-s + (−2.64 + 2.00i)16-s + (−1.03 + 0.227i)17-s + ⋯
L(s)  = 1  + (−0.237 − 1.44i)2-s + (0.154 + 0.556i)3-s + (−1.08 + 0.366i)4-s + (0.420 − 0.619i)5-s + (0.768 − 0.355i)6-s + (1.77 + 0.391i)7-s + (0.101 + 0.192i)8-s + (−0.285 + 0.171i)9-s + (−0.996 − 0.460i)10-s + (−0.427 − 0.325i)11-s + (−0.372 − 0.548i)12-s + (−0.904 − 0.544i)13-s + (0.144 − 2.66i)14-s + (0.409 + 0.138i)15-s + (−0.660 + 0.502i)16-s + (−0.250 + 0.0551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810509 - 0.909473i\)
\(L(\frac12)\) \(\approx\) \(0.810509 - 0.909473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.267 - 0.963i)T \)
59 \( 1 + (6.99 + 3.16i)T \)
good2 \( 1 + (0.335 + 2.04i)T + (-1.89 + 0.638i)T^{2} \)
5 \( 1 + (-0.939 + 1.38i)T + (-1.85 - 4.64i)T^{2} \)
7 \( 1 + (-4.70 - 1.03i)T + (6.35 + 2.93i)T^{2} \)
11 \( 1 + (1.41 + 1.07i)T + (2.94 + 10.5i)T^{2} \)
13 \( 1 + (3.26 + 1.96i)T + (6.08 + 11.4i)T^{2} \)
17 \( 1 + (1.03 - 0.227i)T + (15.4 - 7.13i)T^{2} \)
19 \( 1 + (-3.52 - 0.383i)T + (18.5 + 4.08i)T^{2} \)
23 \( 1 + (-2.75 + 3.24i)T + (-3.72 - 22.6i)T^{2} \)
29 \( 1 + (1.14 - 6.96i)T + (-27.4 - 9.25i)T^{2} \)
31 \( 1 + (7.28 - 0.792i)T + (30.2 - 6.66i)T^{2} \)
37 \( 1 + (-1.92 + 3.63i)T + (-20.7 - 30.6i)T^{2} \)
41 \( 1 + (-4.11 - 4.84i)T + (-6.63 + 40.4i)T^{2} \)
43 \( 1 + (5.34 - 4.06i)T + (11.5 - 41.4i)T^{2} \)
47 \( 1 + (-3.53 - 5.22i)T + (-17.3 + 43.6i)T^{2} \)
53 \( 1 + (7.36 - 3.40i)T + (34.3 - 40.3i)T^{2} \)
61 \( 1 + (1.03 + 6.29i)T + (-57.8 + 19.4i)T^{2} \)
67 \( 1 + (-5.07 - 9.56i)T + (-37.5 + 55.4i)T^{2} \)
71 \( 1 + (7.41 + 10.9i)T + (-26.2 + 65.9i)T^{2} \)
73 \( 1 + (0.132 - 2.44i)T + (-72.5 - 7.89i)T^{2} \)
79 \( 1 + (-4.04 + 14.5i)T + (-67.6 - 40.7i)T^{2} \)
83 \( 1 + (10.8 - 10.3i)T + (4.49 - 82.8i)T^{2} \)
89 \( 1 + (-0.177 + 1.08i)T + (-84.3 - 28.4i)T^{2} \)
97 \( 1 + (-0.525 - 9.69i)T + (-96.4 + 10.4i)T^{2} \)
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

−12.27605691810971723556553276099, −11.13888638327683751296235005496, −10.77403721238174381707002687460, −9.490042332092662784985019204444, −8.824359598088697055466236335193, −7.72482674598771653726715764717, −5.35849353619970803892147036714, −4.65201673745463351170716743084, −2.92693858040266984397980484194, −1.56231304305787179874822128324, 2.17405457712729405352603759032, 4.71182515970842862430012690508, 5.68264100702634578397184381354, 7.13031569216565990697910787485, 7.47391181943569206089607667172, 8.459795628599034679380604719594, 9.653526849930033274522022824546, 11.04882472685301939819150776636, 11.88462706992274407472130483020, 13.51584325131699404689004094438

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