↑

Properties

Label 2-162-9.7-c7-0-21
Degree $2$
Conductor $162$
Sign $-0.939 + 0.342i$
Analytic cond. $50.6063$
Root an. cond. $7.11381$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)2-s + (−31.9 − 55.4i)4-s + (105 + 181. i)5-s + (−508 + 879. i)7-s − 511.·8-s + 1.68e3·10-s + (−546 + 945. i)11-s + (−691 − 1.19e3i)13-s + (4.06e3 + 7.03e3i)14-s + (−2.04e3 + 3.54e3i)16-s + 1.47e4·17-s − 3.99e4·19-s + (6.72e3 − 1.16e4i)20-s + (4.36e3 + 7.56e3i)22-s + (−3.43e4 − 5.95e4i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.375 + 0.650i)5-s + (−0.559 + 0.969i)7-s − 0.353·8-s + 0.531·10-s + (−0.123 + 0.214i)11-s + (−0.0872 − 0.151i)13-s + (0.395 + 0.685i)14-s + (−0.125 + 0.216i)16-s + 0.725·17-s − 1.33·19-s + (0.187 − 0.325i)20-s + (0.0874 + 0.151i)22-s + (−0.588 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(50.6063\)
Root analytic conductor: \(7.11381\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :7/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.7177402795\)
\(L(\frac12)\) \(\approx\) \(0.7177402795\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 6.92i)T \)
3 \( 1 \)
good5 \( 1 + (-105 - 181. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (508 - 879. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (546 - 945. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (691 + 1.19e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 - 1.47e4T + 4.10e8T^{2} \)
19 \( 1 + 3.99e4T + 8.93e8T^{2} \)
23 \( 1 + (3.43e4 + 5.95e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-5.12e4 + 8.88e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (1.13e5 + 1.97e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 - 1.60e5T + 9.49e10T^{2} \)
41 \( 1 + (5.42e3 + 9.38e3i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (-3.15e5 + 5.46e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (2.36e5 - 4.09e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + 1.49e6T + 1.17e12T^{2} \)
59 \( 1 + (1.32e6 + 2.28e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (4.13e5 - 7.16e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-6.30e4 - 1.09e5i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 1.41e6T + 9.09e12T^{2} \)
73 \( 1 - 9.80e5T + 1.10e13T^{2} \)
79 \( 1 + (-1.78e6 + 3.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (2.83e6 - 4.91e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + 1.19e7T + 4.42e13T^{2} \)
97 \( 1 + (4.34e6 - 7.51e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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

−11.06787318355986964097171174062, −10.18899675184351352853608829798, −9.349678879799478911701259624787, −8.118665624761470450068917685764, −6.48516845401651517589623430707, −5.78086472440448823956642656773, −4.30458204941907521524887790694, −2.86361606282305645118690798344, −2.10434678322277747658240576645, −0.16362937463392281351662987648, 1.34819969205301651089558808919, 3.28627334425985638000464321709, 4.44166331907479062758973240146, 5.59776990575650498259692524053, 6.68285425174787471801328566547, 7.71774133683985805629707949672, 8.856416775850293193499677398173, 9.861474803056141706691768079576, 10.92875447280971535719004613922, 12.38590282405699067310956022452

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