↑

Properties

Label 2-640-80.67-c1-0-4
Degree $2$
Conductor $640$
Sign $-0.522 - 0.852i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  + 2.96i·3-s + (2.22 − 0.177i)5-s + (0.115 + 0.115i)7-s − 5.79·9-s + (2.95 + 2.95i)11-s − 1.55·13-s + (0.525 + 6.61i)15-s + (0.299 + 0.299i)17-s + (2.26 + 2.26i)19-s + (−0.341 + 0.341i)21-s + (−4.14 + 4.14i)23-s + (4.93 − 0.790i)25-s − 8.28i·27-s + (0.289 − 0.289i)29-s − 4.18i·31-s + ⋯
L(s)  = 1  + 1.71i·3-s + (0.996 − 0.0793i)5-s + (0.0435 + 0.0435i)7-s − 1.93·9-s + (0.892 + 0.892i)11-s − 0.432·13-s + (0.135 + 1.70i)15-s + (0.0726 + 0.0726i)17-s + (0.519 + 0.519i)19-s + (−0.0744 + 0.0744i)21-s + (−0.864 + 0.864i)23-s + (0.987 − 0.158i)25-s − 1.59i·27-s + (0.0537 − 0.0537i)29-s − 0.751i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828903 + 1.47953i\)
\(L(\frac12)\) \(\approx\) \(0.828903 + 1.47953i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 0.177i)T \)
good3 \( 1 - 2.96iT - 3T^{2} \)
7 \( 1 + (-0.115 - 0.115i)T + 7iT^{2} \)
11 \( 1 + (-2.95 - 2.95i)T + 11iT^{2} \)
13 \( 1 + 1.55T + 13T^{2} \)
17 \( 1 + (-0.299 - 0.299i)T + 17iT^{2} \)
19 \( 1 + (-2.26 - 2.26i)T + 19iT^{2} \)
23 \( 1 + (4.14 - 4.14i)T - 23iT^{2} \)
29 \( 1 + (-0.289 + 0.289i)T - 29iT^{2} \)
31 \( 1 + 4.18iT - 31T^{2} \)
37 \( 1 + 1.63T + 37T^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 + 6.72T + 43T^{2} \)
47 \( 1 + (-4.38 + 4.38i)T - 47iT^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (1.63 - 1.63i)T - 59iT^{2} \)
61 \( 1 + (-1.23 - 1.23i)T + 61iT^{2} \)
67 \( 1 - 2.49T + 67T^{2} \)
71 \( 1 + 8.00T + 71T^{2} \)
73 \( 1 + (-1.12 - 1.12i)T + 73iT^{2} \)
79 \( 1 - 3.62T + 79T^{2} \)
83 \( 1 + 1.62iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + (-9.69 - 9.69i)T + 97iT^{2} \)
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

−10.44836973163608785784756317662, −9.957519570206403976913430925411, −9.413796872437587977320563825933, −8.705670339153310880888019330778, −7.32980774152442018261121110532, −6.03529620799923368551550380480, −5.29426285854525969780205646036, −4.38390993579675627029574073792, −3.47397366445321092779887262830, −2.00648120568795610119620278853, 0.975411902236036874611344065725, 2.07030561066990773723757376529, 3.15467841792506857280605836279, 5.03286041593444351916797598483, 6.20963270800685024949273075237, 6.51299412943595994407369697292, 7.52208888461460153299606385393, 8.463280445116229190642355905652, 9.217364804680957544531558791547, 10.30956880461378799541064789660

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