↑

Properties

Label 2-161-161.45-c1-0-9
Degree $2$
Conductor $161$
Sign $0.932 + 0.361i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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  + (0.292 + 0.506i)2-s + (0.510 + 0.294i)3-s + (0.829 − 1.43i)4-s + (−1.76 − 3.05i)5-s + 0.344i·6-s + (2.22 + 1.43i)7-s + 2.13·8-s + (−1.32 − 2.29i)9-s + (1.03 − 1.78i)10-s + (−0.243 − 0.140i)11-s + (0.846 − 0.488i)12-s + 6.20i·13-s + (−0.0792 + 1.54i)14-s − 2.08i·15-s + (−1.03 − 1.79i)16-s + (−1.89 + 3.28i)17-s + ⋯
L(s)  = 1  + (0.206 + 0.357i)2-s + (0.294 + 0.170i)3-s + (0.414 − 0.718i)4-s + (−0.790 − 1.36i)5-s + 0.140i·6-s + (0.839 + 0.543i)7-s + 0.755·8-s + (−0.442 − 0.765i)9-s + (0.326 − 0.565i)10-s + (−0.0733 − 0.0423i)11-s + (0.244 − 0.141i)12-s + 1.72i·13-s + (−0.0211 + 0.412i)14-s − 0.537i·15-s + (−0.258 − 0.447i)16-s + (−0.460 + 0.797i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37428 - 0.257465i\)
\(L(\frac12)\) \(\approx\) \(1.37428 - 0.257465i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.22 - 1.43i)T \)
23 \( 1 + (-4.78 + 0.372i)T \)
good2 \( 1 + (-0.292 - 0.506i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.510 - 0.294i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.76 + 3.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.243 + 0.140i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.20iT - 13T^{2} \)
17 \( 1 + (1.89 - 3.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 2.64T + 29T^{2} \)
31 \( 1 + (4.59 + 2.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.645 + 0.372i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.20iT - 41T^{2} \)
43 \( 1 - 7.90iT - 43T^{2} \)
47 \( 1 + (5.41 - 3.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.15 - 0.664i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.11 - 1.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.17 + 8.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.69 + 5.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.77T + 71T^{2} \)
73 \( 1 + (-6.45 - 3.72i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.19 + 3.57i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (2.79 + 4.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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

−12.73521310062000690000805037602, −11.73193389503808730690571721246, −11.16824333521101360890288687304, −9.397697759433678869627377289637, −8.791245195421682891327452722916, −7.70358020967331542885638566115, −6.25330695811111162206507541230, −5.06751426054817791512220002540, −4.11115985068732770952850420502, −1.63170443547962894657868202379, 2.61020537051098237097086166868, 3.43478732647794967673011897909, 5.04192196988942998692936817554, 7.16508222285407294352513431529, 7.49858012139684080445381488060, 8.445876357314696281722681890122, 10.51633810724519342367937496235, 11.03410119640308966418557875624, 11.63260226006320875218776211353, 13.02594940495310867136902761507

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