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Properties

Label 2-640-80.43-c1-0-0
Degree $2$
Conductor $640$
Sign $-0.270 - 0.962i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.55i·3-s + (−1.66 − 1.49i)5-s + (−2.40 + 2.40i)7-s − 3.51·9-s + (−2.67 + 2.67i)11-s + 2.40·13-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 + 1.30i·27-s + (−3.95 − 3.95i)29-s + 1.65i·31-s + ⋯
L(s)  = 1  − 1.47i·3-s + (−0.745 − 0.666i)5-s + (−0.908 + 0.908i)7-s − 1.17·9-s + (−0.807 + 0.807i)11-s + 0.666·13-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 + 0.250i·27-s + (−0.734 − 0.734i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.270 - 0.962i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.270 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0249050 + 0.0328690i\)
\(L(\frac12)\) \(\approx\) \(0.0249050 + 0.0328690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 1.49i)T \)
good3 \( 1 + 2.55iT - 3T^{2} \)
7 \( 1 + (2.40 - 2.40i)T - 7iT^{2} \)
11 \( 1 + (2.67 - 2.67i)T - 11iT^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
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 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 + 3.84T + 43T^{2} \)
47 \( 1 + (-2.15 - 2.15i)T + 47iT^{2} \)
53 \( 1 - 1.29iT - 53T^{2} \)
59 \( 1 + (5.29 + 5.29i)T + 59iT^{2} \)
61 \( 1 + (10.2 - 10.2i)T - 61iT^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (-9.99 + 9.99i)T - 73iT^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 + 5.00i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.04917929450475844574914084728, −9.839249200667782016230065634112, −8.827217832030726995467649792771, −8.083382773649673445515780439660, −7.42339146883486251863451665110, −6.37836118395123933298242342287, −5.68489747486086689614659289139, −4.30473963905593425706884196971, −2.86984700045289212531464013003, −1.70368016895609084954475262836, 0.02155746533670264461171777044, 3.03747040653858263992373678129, 3.65470078611020024419502246352, 4.43552446004909179619474572258, 5.69360705895506183510070125384, 6.72715965458933113825026536408, 7.73313792446257093323431974709, 8.734571333232150992304591373180, 9.636718746017868220452190292731, 10.60433054717658225286205958323

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