↑

Properties

Label 2-320-40.3-c1-0-5
Degree $2$
Conductor $320$
Sign $0.572 - 0.819i$
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  + (2.18 + 2.18i)3-s + (−0.456 − 2.18i)5-s + (1.79 + 1.79i)7-s + 6.58i·9-s − 0.913·11-s + (−1.73 + 1.73i)13-s + (3.79 − 5.79i)15-s + (3 − 3i)17-s − 3.46i·19-s + 7.84i·21-s + (3.79 − 3.79i)23-s + (−4.58 + 1.99i)25-s + (−7.84 + 7.84i)27-s − 5.29·29-s + 7.58i·31-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)3-s + (−0.204 − 0.978i)5-s + (0.677 + 0.677i)7-s + 2.19i·9-s − 0.275·11-s + (−0.480 + 0.480i)13-s + (0.978 − 1.49i)15-s + (0.727 − 0.727i)17-s − 0.794i·19-s + 1.71i·21-s + (0.790 − 0.790i)23-s + (−0.916 + 0.399i)25-s + (−1.50 + 1.50i)27-s − 0.982·29-s + 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.572 - 0.819i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.572 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70330 + 0.888035i\)
\(L(\frac12)\) \(\approx\) \(1.70330 + 0.888035i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.456 + 2.18i)T \)
good3 \( 1 + (-2.18 - 2.18i)T + 3iT^{2} \)
7 \( 1 + (-1.79 - 1.79i)T + 7iT^{2} \)
11 \( 1 + 0.913T + 11T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-3.79 + 3.79i)T - 23iT^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 - 7.58iT - 31T^{2} \)
37 \( 1 + (5.19 + 5.19i)T + 37iT^{2} \)
41 \( 1 - 1.58T + 41T^{2} \)
43 \( 1 + (-0.361 - 0.361i)T + 43iT^{2} \)
47 \( 1 + (3.79 + 3.79i)T + 47iT^{2} \)
53 \( 1 + (-2.64 + 2.64i)T - 53iT^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 + 6.20iT - 61T^{2} \)
67 \( 1 + (0.361 - 0.361i)T - 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (8.58 + 8.58i)T + 73iT^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (-3.10 - 3.10i)T + 83iT^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + (8.58 - 8.58i)T - 97iT^{2} \)
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

−11.71340387028036167200249467648, −10.68969510583685538307049625836, −9.581870012834576800691672128071, −8.967099869425756385047019180556, −8.395930661539507612973823591481, −7.36951095107232986622605229937, −5.13438410011949035827611955586, −4.84907233032674634405910677312, −3.50072910890815620855250866064, −2.20688305333237280610987585714, 1.56856732259898553235872835770, 2.87327222323351124839965407479, 3.85544204918237329699425067864, 5.84367522706782448088251062787, 7.11370883195068802927833057079, 7.66980666709433218748833726674, 8.202299008647053676571322273917, 9.568337619037770038401931490101, 10.55159755226388338682878342731, 11.60673539132555124111742365151

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