↑

Properties

Label 2-162-81.16-c1-0-8
Degree $2$
Conductor $162$
Sign $0.984 - 0.177i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
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  + (0.973 + 0.230i)2-s + (1.70 + 0.290i)3-s + (0.893 + 0.448i)4-s + (0.537 − 0.721i)5-s + (1.59 + 0.676i)6-s + (−3.95 − 2.60i)7-s + (0.766 + 0.642i)8-s + (2.83 + 0.992i)9-s + (0.689 − 0.578i)10-s + (−4.21 + 0.492i)11-s + (1.39 + 1.02i)12-s + (−1.75 + 5.86i)13-s + (−3.24 − 3.44i)14-s + (1.12 − 1.07i)15-s + (0.597 + 0.802i)16-s + (−0.432 − 2.45i)17-s + ⋯
L(s)  = 1  + (0.688 + 0.163i)2-s + (0.985 + 0.167i)3-s + (0.446 + 0.224i)4-s + (0.240 − 0.322i)5-s + (0.650 + 0.276i)6-s + (−1.49 − 0.983i)7-s + (0.270 + 0.227i)8-s + (0.943 + 0.330i)9-s + (0.217 − 0.182i)10-s + (−1.26 + 0.148i)11-s + (0.402 + 0.296i)12-s + (−0.487 + 1.62i)13-s + (−0.868 − 0.920i)14-s + (0.291 − 0.277i)15-s + (0.149 + 0.200i)16-s + (−0.104 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.984 - 0.177i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.984 - 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91919 + 0.171820i\)
\(L(\frac12)\) \(\approx\) \(1.91919 + 0.171820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.973 - 0.230i)T \)
3 \( 1 + (-1.70 - 0.290i)T \)
good5 \( 1 + (-0.537 + 0.721i)T + (-1.43 - 4.78i)T^{2} \)
7 \( 1 + (3.95 + 2.60i)T + (2.77 + 6.42i)T^{2} \)
11 \( 1 + (4.21 - 0.492i)T + (10.7 - 2.53i)T^{2} \)
13 \( 1 + (1.75 - 5.86i)T + (-10.8 - 7.14i)T^{2} \)
17 \( 1 + (0.432 + 2.45i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (0.284 - 1.61i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-6.35 + 4.17i)T + (9.10 - 21.1i)T^{2} \)
29 \( 1 + (-2.09 + 2.21i)T + (-1.68 - 28.9i)T^{2} \)
31 \( 1 + (0.107 + 1.84i)T + (-30.7 + 3.59i)T^{2} \)
37 \( 1 + (8.42 + 3.06i)T + (28.3 + 23.7i)T^{2} \)
41 \( 1 + (-5.04 + 1.19i)T + (36.6 - 18.4i)T^{2} \)
43 \( 1 + (-1.06 + 2.46i)T + (-29.5 - 31.2i)T^{2} \)
47 \( 1 + (-0.486 + 8.35i)T + (-46.6 - 5.45i)T^{2} \)
53 \( 1 + (1.11 + 1.93i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.71 + 0.433i)T + (57.4 + 13.6i)T^{2} \)
61 \( 1 + (-2.81 + 1.41i)T + (36.4 - 48.9i)T^{2} \)
67 \( 1 + (-3.28 - 3.48i)T + (-3.89 + 66.8i)T^{2} \)
71 \( 1 + (3.18 - 2.67i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.09 - 0.922i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (5.03 + 1.19i)T + (70.5 + 35.4i)T^{2} \)
83 \( 1 + (2.63 + 0.624i)T + (74.1 + 37.2i)T^{2} \)
89 \( 1 + (-12.5 - 10.5i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.79 - 3.75i)T + (-27.8 + 92.9i)T^{2} \)
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

−13.19134611871119355281362995050, −12.39165404853972178686072889063, −10.71897108539107005664582091102, −9.802519275269370745863797597508, −8.923740243562694930650066007403, −7.35911102685172985824918700081, −6.76930066405254153411832754788, −4.95543185954429229297530967145, −3.80400254564471347820740126459, −2.54352155780279284700628637825, 2.76745412678928797247351510067, 3.10460446898260012657750010895, 5.20912162212214439184039484586, 6.34470074635749085602795074163, 7.54684454753043927427250536867, 8.777445811289761209946368885212, 9.922171153356912434171272250920, 10.63531137735042718010294738982, 12.52773176027738623641992322486, 12.79463850023032950976361596949

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