↑

Properties

Label 2-640-80.27-c1-0-0
Degree $2$
Conductor $640$
Sign $0.296 - 0.955i$
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  − 2.55·3-s + (−1.49 − 1.66i)5-s + (−2.40 − 2.40i)7-s + 3.51·9-s + (−2.67 + 2.67i)11-s − 2.40i·13-s + (3.80 + 4.25i)15-s + (−0.0750 − 0.0750i)17-s + (2.67 − 2.67i)19-s + (6.13 + 6.13i)21-s + (−2.12 + 2.12i)23-s + (−0.553 + 4.96i)25-s − 1.30·27-s + (3.95 + 3.95i)29-s + 1.65i·31-s + ⋯
L(s)  = 1  − 1.47·3-s + (−0.666 − 0.745i)5-s + (−0.908 − 0.908i)7-s + 1.17·9-s + (−0.807 + 0.807i)11-s − 0.666i·13-s + (0.982 + 1.09i)15-s + (−0.0182 − 0.0182i)17-s + (0.613 − 0.613i)19-s + (1.33 + 1.33i)21-s + (−0.442 + 0.442i)23-s + (−0.110 + 0.993i)25-s − 0.250·27-s + (0.734 + 0.734i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.296 - 0.955i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208738 + 0.153831i\)
\(L(\frac12)\) \(\approx\) \(0.208738 + 0.153831i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 + 1.66i)T \)
good3 \( 1 + 2.55T + 3T^{2} \)
7 \( 1 + (2.40 + 2.40i)T + 7iT^{2} \)
11 \( 1 + (2.67 - 2.67i)T - 11iT^{2} \)
13 \( 1 + 2.40iT - 13T^{2} \)
17 \( 1 + (0.0750 + 0.0750i)T + 17iT^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 + (2.12 - 2.12i)T - 23iT^{2} \)
29 \( 1 + (-3.95 - 3.95i)T + 29iT^{2} \)
31 \( 1 - 1.65iT - 31T^{2} \)
37 \( 1 - 2.53iT - 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 - 3.84iT - 43T^{2} \)
47 \( 1 + (2.15 - 2.15i)T - 47iT^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 + (-5.29 - 5.29i)T + 59iT^{2} \)
61 \( 1 + (10.2 - 10.2i)T - 61iT^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (9.99 + 9.99i)T + 73iT^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 - 5.00i)T + 97iT^{2} \)
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

−10.63984300112145768213251516457, −10.21848363724163678430615518740, −9.201334510230246299084344773515, −7.82777399240266534818730784771, −7.18136931853773359258607929544, −6.22153844547314234738130705983, −5.13723217543686857495368624796, −4.57111080548510746761526099161, −3.26512575516190366364708646520, −0.952086599249140798175243244358, 0.22287894631369152232012482650, 2.60336004086023131712467853939, 3.78739751848026428847992966112, 5.11177570696675334209958601316, 6.05290997971036617342347237297, 6.46904199060639052425356127093, 7.58697655054817430238108975878, 8.629366153489603635661421806826, 9.855252754166606358583481196067, 10.54175978697389346764006032285

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