↑

Properties

Label 2-320-20.3-c3-0-25
Degree $2$
Conductor $320$
Sign $0.0323 + 0.999i$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.99 − 2.99i)3-s + (−9.31 + 6.18i)5-s + (6.68 + 6.68i)7-s + 9.09i·9-s − 63.1i·11-s + (−28.2 − 28.2i)13-s + (−9.37 + 46.3i)15-s + (30.3 − 30.3i)17-s + 22.2·19-s + 40.0·21-s + (86.5 − 86.5i)23-s + (48.5 − 115. i)25-s + (107. + 107. i)27-s − 188. i·29-s − 71.2i·31-s + ⋯
L(s)  = 1  + (0.575 − 0.575i)3-s + (−0.833 + 0.552i)5-s + (0.361 + 0.361i)7-s + 0.336i·9-s − 1.73i·11-s + (−0.602 − 0.602i)13-s + (−0.161 + 0.798i)15-s + (0.433 − 0.433i)17-s + 0.268·19-s + 0.415·21-s + (0.784 − 0.784i)23-s + (0.388 − 0.921i)25-s + (0.769 + 0.769i)27-s − 1.20i·29-s − 0.413i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0323 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0323 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.0323 + 0.999i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ 0.0323 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.666657006\)
\(L(\frac12)\) \(\approx\) \(1.666657006\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.31 - 6.18i)T \)
good3 \( 1 + (-2.99 + 2.99i)T - 27iT^{2} \)
7 \( 1 + (-6.68 - 6.68i)T + 343iT^{2} \)
11 \( 1 + 63.1iT - 1.33e3T^{2} \)
13 \( 1 + (28.2 + 28.2i)T + 2.19e3iT^{2} \)
17 \( 1 + (-30.3 + 30.3i)T - 4.91e3iT^{2} \)
19 \( 1 - 22.2T + 6.85e3T^{2} \)
23 \( 1 + (-86.5 + 86.5i)T - 1.21e4iT^{2} \)
29 \( 1 + 188. iT - 2.43e4T^{2} \)
31 \( 1 + 71.2iT - 2.97e4T^{2} \)
37 \( 1 + (-217. + 217. i)T - 5.06e4iT^{2} \)
41 \( 1 + 4.57T + 6.89e4T^{2} \)
43 \( 1 + (392. - 392. i)T - 7.95e4iT^{2} \)
47 \( 1 + (280. + 280. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-82.3 - 82.3i)T + 1.48e5iT^{2} \)
59 \( 1 - 794.T + 2.05e5T^{2} \)
61 \( 1 + 714.T + 2.26e5T^{2} \)
67 \( 1 + (373. + 373. i)T + 3.00e5iT^{2} \)
71 \( 1 - 528. iT - 3.57e5T^{2} \)
73 \( 1 + (531. + 531. i)T + 3.89e5iT^{2} \)
79 \( 1 - 349.T + 4.93e5T^{2} \)
83 \( 1 + (112. - 112. i)T - 5.71e5iT^{2} \)
89 \( 1 - 424. iT - 7.04e5T^{2} \)
97 \( 1 + (-501. + 501. i)T - 9.12e5iT^{2} \)
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

−11.19954004155837990538994716074, −10.15153597564714005135544031679, −8.716584946401004802422216540765, −8.062686571228321712966629572589, −7.41379317425411171736632012187, −6.16103733199966248665703016700, −4.92660767516750792228179317491, −3.33994475103438202572472442416, −2.53201181269022478508767111340, −0.59804987898848162341580696175, 1.46274032360310808833128618139, 3.26659098404103474494012704845, 4.35627389746065804691170118696, 4.98418475175561916833030590596, 6.91469372207404813672126294327, 7.62082837565989633275180312726, 8.722857909785787805990401750252, 9.521536485635321786213262663717, 10.28132795228827034354798803479, 11.59273154767158883203945517769

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