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Properties

Label 2-320-80.3-c1-0-2
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

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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: $\chi_{320} (303, \cdot )$
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 \( 1 + (7.15 - 7.15i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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