↑

Properties

Label 2-320-80.27-c1-0-8
Degree $2$
Conductor $320$
Sign $-0.201 + 0.979i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
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  + 0.496·3-s + (−2.00 − 0.987i)5-s + (−1.55 − 1.55i)7-s − 2.75·9-s + (4.19 − 4.19i)11-s − 5.09i·13-s + (−0.996 − 0.490i)15-s + (0.213 + 0.213i)17-s + (−0.844 + 0.844i)19-s + (−0.771 − 0.771i)21-s + (−1.70 + 1.70i)23-s + (3.05 + 3.96i)25-s − 2.85·27-s + (2.24 + 2.24i)29-s − 0.818i·31-s + ⋯
L(s)  = 1  + 0.286·3-s + (−0.897 − 0.441i)5-s + (−0.587 − 0.587i)7-s − 0.917·9-s + (1.26 − 1.26i)11-s − 1.41i·13-s + (−0.257 − 0.126i)15-s + (0.0517 + 0.0517i)17-s + (−0.193 + 0.193i)19-s + (−0.168 − 0.168i)21-s + (−0.356 + 0.356i)23-s + (0.610 + 0.792i)25-s − 0.549·27-s + (0.417 + 0.417i)29-s − 0.146i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.201 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597192 - 0.732537i\)
\(L(\frac12)\) \(\approx\) \(0.597192 - 0.732537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.00 + 0.987i)T \)
good3 \( 1 - 0.496T + 3T^{2} \)
7 \( 1 + (1.55 + 1.55i)T + 7iT^{2} \)
11 \( 1 + (-4.19 + 4.19i)T - 11iT^{2} \)
13 \( 1 + 5.09iT - 13T^{2} \)
17 \( 1 + (-0.213 - 0.213i)T + 17iT^{2} \)
19 \( 1 + (0.844 - 0.844i)T - 19iT^{2} \)
23 \( 1 + (1.70 - 1.70i)T - 23iT^{2} \)
29 \( 1 + (-2.24 - 2.24i)T + 29iT^{2} \)
31 \( 1 + 0.818iT - 31T^{2} \)
37 \( 1 + 5.12iT - 37T^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 + 4.49iT - 43T^{2} \)
47 \( 1 + (4.29 - 4.29i)T - 47iT^{2} \)
53 \( 1 + 1.00T + 53T^{2} \)
59 \( 1 + (-7.65 - 7.65i)T + 59iT^{2} \)
61 \( 1 + (1.90 - 1.90i)T - 61iT^{2} \)
67 \( 1 + 11.0iT - 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-2.70 - 2.70i)T + 73iT^{2} \)
79 \( 1 - 8.32T + 79T^{2} \)
83 \( 1 - 9.17T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
97 \( 1 + (7.15 + 7.15i)T + 97iT^{2} \)
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

−11.38986102714329265379989019501, −10.59536950528408499669089704318, −9.306482053126036079947028145974, −8.474449358014053243929034070075, −7.78607192328009456837364950327, −6.47318955976517642547966272383, −5.44542106881997201444486674469, −3.82877346014283486981423472588, −3.21324158233275002904770497356, −0.66052134525692330912898216396, 2.27271912663340714442667386554, 3.61970901480538938655433858519, 4.62470982294945855221652965536, 6.36524222098328431088674262180, 6.92042711455089019835838231761, 8.216641976795823615001279998489, 9.097622804459888678845348105417, 9.831648003183464382594110312480, 11.26577900167434066739212068887, 11.87402069089924580330445236688

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