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Properties

Label 2-640-80.67-c1-0-15
Degree $2$
Conductor $640$
Sign $-0.884 + 0.467i$
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  − 1.39i·3-s + (−0.535 + 2.17i)5-s + (−2.13 − 2.13i)7-s + 1.05·9-s + (−2.17 − 2.17i)11-s − 1.54·13-s + (3.02 + 0.745i)15-s + (−3.86 − 3.86i)17-s + (0.0136 + 0.0136i)19-s + (−2.97 + 2.97i)21-s + (−3.15 + 3.15i)23-s + (−4.42 − 2.32i)25-s − 5.65i·27-s + (3.33 − 3.33i)29-s − 8.92i·31-s + ⋯
L(s)  = 1  − 0.804i·3-s + (−0.239 + 0.970i)5-s + (−0.806 − 0.806i)7-s + 0.353·9-s + (−0.654 − 0.654i)11-s − 0.428·13-s + (0.780 + 0.192i)15-s + (−0.937 − 0.937i)17-s + (0.00313 + 0.00313i)19-s + (−0.648 + 0.648i)21-s + (−0.657 + 0.657i)23-s + (−0.885 − 0.464i)25-s − 1.08i·27-s + (0.619 − 0.619i)29-s − 1.60i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.884 + 0.467i$
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: \((2,\ 640,\ (\ :1/2),\ -0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156095 - 0.629697i\)
\(L(\frac12)\) \(\approx\) \(0.156095 - 0.629697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.535 - 2.17i)T \)
good3 \( 1 + 1.39iT - 3T^{2} \)
7 \( 1 + (2.13 + 2.13i)T + 7iT^{2} \)
11 \( 1 + (2.17 + 2.17i)T + 11iT^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 + (3.86 + 3.86i)T + 17iT^{2} \)
19 \( 1 + (-0.0136 - 0.0136i)T + 19iT^{2} \)
23 \( 1 + (3.15 - 3.15i)T - 23iT^{2} \)
29 \( 1 + (-3.33 + 3.33i)T - 29iT^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 2.02T + 43T^{2} \)
47 \( 1 + (3.34 - 3.34i)T - 47iT^{2} \)
53 \( 1 + 7.30iT - 53T^{2} \)
59 \( 1 + (3.52 - 3.52i)T - 59iT^{2} \)
61 \( 1 + (1.41 + 1.41i)T + 61iT^{2} \)
67 \( 1 + 0.748T + 67T^{2} \)
71 \( 1 + 0.269T + 71T^{2} \)
73 \( 1 + (0.811 + 0.811i)T + 73iT^{2} \)
79 \( 1 - 2.80T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-6.33 - 6.33i)T + 97iT^{2} \)
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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

−10.14076781232573608068781791786, −9.628631039269295150041267925099, −8.092900078847511994115692283131, −7.44192485958160270873803721318, −6.75318400863341211098721725862, −6.05067124378634780010532257598, −4.47504457626739637308014254883, −3.32996360132752220141600802613, −2.28665586216818033584261691685, −0.33059600635445920082049906134, 2.01737750290950670755329915064, 3.50275217434407591653345803070, 4.58241746940784323039349272261, 5.19665118830480250477605456228, 6.38517611691212366649653653626, 7.47345197305982606135247672166, 8.775153319423575261403395319668, 9.013762245567135512844470275805, 10.20511126921125809035031448259, 10.52338966119361725533841573374

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