↑

Properties

Label 2-171-171.121-c1-0-17
Degree $2$
Conductor $171$
Sign $-0.977 - 0.212i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
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  + (1.01 − 1.74i)2-s + (−1.70 − 0.311i)3-s + (−1.04 − 1.80i)4-s − 4.18·5-s + (−2.26 + 2.66i)6-s + (−0.976 − 1.69i)7-s − 0.164·8-s + (2.80 + 1.06i)9-s + (−4.22 + 7.32i)10-s + (−0.669 − 1.15i)11-s + (1.21 + 3.39i)12-s + (−0.975 − 1.68i)13-s − 3.94·14-s + (7.13 + 1.30i)15-s + (1.91 − 3.31i)16-s + (−3.34 − 5.78i)17-s + ⋯
L(s)  = 1  + (0.714 − 1.23i)2-s + (−0.983 − 0.179i)3-s + (−0.520 − 0.901i)4-s − 1.87·5-s + (−0.925 + 1.08i)6-s + (−0.368 − 0.639i)7-s − 0.0583·8-s + (0.935 + 0.354i)9-s + (−1.33 + 2.31i)10-s + (−0.201 − 0.349i)11-s + (0.349 + 0.980i)12-s + (−0.270 − 0.468i)13-s − 1.05·14-s + (1.84 + 0.336i)15-s + (0.478 − 0.829i)16-s + (−0.810 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.977 - 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0729137 + 0.678901i\)
\(L(\frac12)\) \(\approx\) \(0.0729137 + 0.678901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.311i)T \)
19 \( 1 + (-4.11 - 1.44i)T \)
good2 \( 1 + (-1.01 + 1.74i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 4.18T + 5T^{2} \)
7 \( 1 + (0.976 + 1.69i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.669 + 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.975 + 1.68i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.34 + 5.78i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.986 - 1.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + (0.385 - 0.668i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.26T + 37T^{2} \)
41 \( 1 - 7.59T + 41T^{2} \)
43 \( 1 + (3.97 - 6.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.10T + 47T^{2} \)
53 \( 1 + (-3.75 + 6.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.01T + 59T^{2} \)
61 \( 1 - 0.332T + 61T^{2} \)
67 \( 1 + (6.45 + 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.81 - 3.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.48 + 4.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.58 - 4.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.30 - 10.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.569 + 0.985i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.87 + 3.25i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.03653587449847624822748283531, −11.32744530001886813099496802387, −10.94966183691660231861118909540, −9.707295044680612506521147155936, −7.74219952109247183554849489239, −7.11129455285726244736030915639, −5.17298410870564508811532078619, −4.22121642600204879555252794815, −3.19912908138102271888578294809, −0.58386483172438276956120003415, 3.88307821445618088766634143341, 4.65930075292953633173201034855, 5.85452019998197360220284591621, 6.96343247048078121931262151175, 7.64237201265233014268276999222, 8.908533964217024877315214320796, 10.62099484359646626586438625170, 11.54263755925810065342328245647, 12.39668874053622867203056530378, 13.10039832777523071385657691750

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