↑

Properties

Label 2-164-164.119-c0-0-0
Degree $2$
Conductor $164$
Sign $0.339 - 0.940i$
Analytic cond. $0.0818466$
Root an. cond. $0.286088$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.5 + 1.53i)5-s + (0.309 + 0.951i)8-s + 9-s + (−0.5 − 1.53i)10-s + (−0.5 + 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)18-s + (1.30 + 0.951i)20-s + (−1.30 − 0.951i)25-s + (0.190 − 0.587i)26-s + (0.618 − 1.90i)29-s + 32-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.5 + 1.53i)5-s + (0.309 + 0.951i)8-s + 9-s + (−0.5 − 1.53i)10-s + (−0.5 + 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)18-s + (1.30 + 0.951i)20-s + (−1.30 − 0.951i)25-s + (0.190 − 0.587i)26-s + (0.618 − 1.90i)29-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.339 - 0.940i$
Analytic conductor: \(0.0818466\)
Root analytic conductor: \(0.286088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :0),\ 0.339 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4532331898\)
\(L(\frac12)\) \(\approx\) \(0.4532331898\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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

−13.66422569665234076379818331246, −11.90096996805712861350720010923, −11.09087056931126405628712169598, −10.14237442661468343105221409066, −9.382633225958132203687287840912, −7.76263992701736982600950997184, −7.15049291583227905713478785896, −6.32044499339019919961226678271, −4.51840247830529123302014758089, −2.56635042137094261371090223348, 1.53451709084940435462792192333, 3.80286650826162953550867050029, 4.91609054497371225976097389874, 6.89428050849731624394414411111, 8.128616982262838387501916121389, 8.742097098320228899416620511390, 9.842889669291685413621983623206, 10.76006302830767359430256150433, 12.09585812256911435691673936200, 12.63758268570595138608685607284

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