↑

Properties

Label 2-320-40.29-c1-0-3
Degree $2$
Conductor $320$
Sign $0.707 - 0.707i$
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  + 2.23·5-s + 4.47i·7-s − 3·9-s + 2i·11-s + 4.47·13-s − 6i·19-s + 4.47i·23-s + 5.00·25-s + 10.0i·35-s + 4.47·37-s + 2·41-s − 6.70·45-s − 13.4i·47-s − 13.0·49-s − 13.4·53-s + ⋯
L(s)  = 1  + 0.999·5-s + 1.69i·7-s − 9-s + 0.603i·11-s + 1.24·13-s − 1.37i·19-s + 0.932i·23-s + 1.00·25-s + 1.69i·35-s + 0.735·37-s + 0.312·41-s − 0.999·45-s − 1.95i·47-s − 1.85·49-s − 1.84·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35648 + 0.561874i\)
\(L(\frac12)\) \(\approx\) \(1.35648 + 0.561874i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
good3 \( 1 + 3T^{2} \)
7 \( 1 - 4.47iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 13.4iT - 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−11.63704196417031549761633179358, −11.01974327805385410250502080945, −9.597647036181392085141917428728, −9.034390091315408450909131343683, −8.271948384891855967724027729889, −6.60472382004018630833946462459, −5.79347639874416465994420150139, −5.07115260460609805077004276843, −3.06608783253236899607480695920, −2.01939718674622824220878174677, 1.20604689492161767630898106379, 3.10570645361687311255346272720, 4.29530307611454374512250789960, 5.83546784057651874510305314798, 6.40560275103942236216451918338, 7.79146860430222018772902531898, 8.663088645955679284362336989407, 9.787350606690287013389780238631, 10.72067677867707085561496307944, 11.12489486516667090859004763075

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