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Properties

Label 2-640-80.43-c1-0-1
Degree $2$
Conductor $640$
Sign $-0.473 - 0.880i$
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  − 2.85i·3-s + (−1.43 + 1.71i)5-s + (−0.458 + 0.458i)7-s − 5.15·9-s + (0.492 − 0.492i)11-s − 4.52·13-s + (4.89 + 4.09i)15-s + (−3.12 + 3.12i)17-s + (−4.04 + 4.04i)19-s + (1.31 + 1.31i)21-s + (−1.80 − 1.80i)23-s + (−0.881 − 4.92i)25-s + 6.15i·27-s + (3.83 + 3.83i)29-s − 0.139i·31-s + ⋯
L(s)  = 1  − 1.64i·3-s + (−0.641 + 0.766i)5-s + (−0.173 + 0.173i)7-s − 1.71·9-s + (0.148 − 0.148i)11-s − 1.25·13-s + (1.26 + 1.05i)15-s + (−0.758 + 0.758i)17-s + (−0.928 + 0.928i)19-s + (0.285 + 0.285i)21-s + (−0.376 − 0.376i)23-s + (−0.176 − 0.984i)25-s + 1.18i·27-s + (0.712 + 0.712i)29-s − 0.0251i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.473 - 0.880i$
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: \((2,\ 640,\ (\ :1/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0312992 + 0.0523896i\)
\(L(\frac12)\) \(\approx\) \(0.0312992 + 0.0523896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 - 1.71i)T \)
good3 \( 1 + 2.85iT - 3T^{2} \)
7 \( 1 + (0.458 - 0.458i)T - 7iT^{2} \)
11 \( 1 + (-0.492 + 0.492i)T - 11iT^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 + (3.12 - 3.12i)T - 17iT^{2} \)
19 \( 1 + (4.04 - 4.04i)T - 19iT^{2} \)
23 \( 1 + (1.80 + 1.80i)T + 23iT^{2} \)
29 \( 1 + (-3.83 - 3.83i)T + 29iT^{2} \)
31 \( 1 + 0.139iT - 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 - 7.49T + 43T^{2} \)
47 \( 1 + (4.14 + 4.14i)T + 47iT^{2} \)
53 \( 1 + 2.75iT - 53T^{2} \)
59 \( 1 + (-3.62 - 3.62i)T + 59iT^{2} \)
61 \( 1 + (3.72 - 3.72i)T - 61iT^{2} \)
67 \( 1 + 3.32T + 67T^{2} \)
71 \( 1 - 1.37T + 71T^{2} \)
73 \( 1 + (2.55 - 2.55i)T - 73iT^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + (4.95 - 4.95i)T - 97iT^{2} \)
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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

−10.95129769302574779226044083605, −10.21134295772557661569642555983, −8.742167889419528651687300447717, −8.072694898357944801786818637181, −7.20824298323255184631209437236, −6.64967459056070143056589989660, −5.82704404878524133404396166639, −4.22649159975326927927998592632, −2.83995396633880773277437724604, −1.89722451801574673008418989551, 0.03083719583292884525059599774, 2.66497495737243226322831327811, 3.97251354737057430550808603110, 4.62967123020531065682169910875, 5.21970242397667340486991579958, 6.70252108124444452774484834573, 7.85468987397970011131832419035, 8.879976360249027599455662028653, 9.420676242938369126737873809356, 10.15853330985561120307780341053

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