↑

Properties

Label 2-164-164.139-c0-0-0
Degree $2$
Conductor $164$
Sign $0.213 + 0.976i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.363i)5-s + (−0.809 + 0.587i)8-s + 9-s + (−0.5 + 0.363i)10-s + (−0.5 + 1.53i)13-s + (0.309 + 0.951i)16-s + (−0.5 + 0.363i)17-s + (0.309 − 0.951i)18-s + (0.190 + 0.587i)20-s + (−0.190 − 0.587i)25-s + (1.30 + 0.951i)26-s + (−1.61 − 1.17i)29-s + 32-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.363i)5-s + (−0.809 + 0.587i)8-s + 9-s + (−0.5 + 0.363i)10-s + (−0.5 + 1.53i)13-s + (0.309 + 0.951i)16-s + (−0.5 + 0.363i)17-s + (0.309 − 0.951i)18-s + (0.190 + 0.587i)20-s + (−0.190 − 0.587i)25-s + (1.30 + 0.951i)26-s + (−1.61 − 1.17i)29-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(0.0818466\)
Root analytic conductor: \(0.286088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :0),\ 0.213 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6730693988\)
\(L(\frac12)\) \(\approx\) \(0.6730693988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 - T^{2} \)
5 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
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

−12.75228088160257476273926155357, −11.87941916981601906468934151191, −11.11454760150778095072215009023, −9.854697830069982905008798806864, −9.155819926315708145919903785952, −7.80494820851369137297052795409, −6.36199730178912880174518923209, −4.66038525491298702689378786165, −3.98801745133347043821825367529, −1.98169852803784718594056350641, 3.29979506319207345038635244485, 4.60635659742548917648301939767, 5.82168624400826362925615891849, 7.26639075452259559707267781377, 7.66943768270509480984516754448, 9.087285068668150553599230386055, 10.16184045521053802360709420229, 11.40534947758035451601893471337, 12.82435061080007572560179045074, 13.07459367230845804192251257685

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