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Properties

Label 2-640-80.3-c1-0-13
Degree $2$
Conductor $640$
Sign $0.859 + 0.510i$
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  + 0.496·3-s + (2.00 − 0.987i)5-s + (1.55 − 1.55i)7-s − 2.75·9-s + (4.19 + 4.19i)11-s − 5.09i·13-s + (0.996 − 0.490i)15-s + (0.213 − 0.213i)17-s + (−0.844 − 0.844i)19-s + (0.771 − 0.771i)21-s + (1.70 + 1.70i)23-s + (3.05 − 3.96i)25-s − 2.85·27-s + (−2.24 + 2.24i)29-s − 0.818i·31-s + ⋯
L(s)  = 1  + 0.286·3-s + (0.897 − 0.441i)5-s + (0.587 − 0.587i)7-s − 0.917·9-s + (1.26 + 1.26i)11-s − 1.41i·13-s + (0.257 − 0.126i)15-s + (0.0517 − 0.0517i)17-s + (−0.193 − 0.193i)19-s + (0.168 − 0.168i)21-s + (0.356 + 0.356i)23-s + (0.610 − 0.792i)25-s − 0.549·27-s + (−0.417 + 0.417i)29-s − 0.146i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.859 + 0.510i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.859 + 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91910 - 0.526766i\)
\(L(\frac12)\) \(\approx\) \(1.91910 - 0.526766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.00 + 0.987i)T \)
good3 \( 1 - 0.496T + 3T^{2} \)
7 \( 1 + (-1.55 + 1.55i)T - 7iT^{2} \)
11 \( 1 + (-4.19 - 4.19i)T + 11iT^{2} \)
13 \( 1 + 5.09iT - 13T^{2} \)
17 \( 1 + (-0.213 + 0.213i)T - 17iT^{2} \)
19 \( 1 + (0.844 + 0.844i)T + 19iT^{2} \)
23 \( 1 + (-1.70 - 1.70i)T + 23iT^{2} \)
29 \( 1 + (2.24 - 2.24i)T - 29iT^{2} \)
31 \( 1 + 0.818iT - 31T^{2} \)
37 \( 1 + 5.12iT - 37T^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 - 4.49iT - 43T^{2} \)
47 \( 1 + (-4.29 - 4.29i)T + 47iT^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 + (-7.65 + 7.65i)T - 59iT^{2} \)
61 \( 1 + (-1.90 - 1.90i)T + 61iT^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (-2.70 + 2.70i)T - 73iT^{2} \)
79 \( 1 + 8.32T + 79T^{2} \)
83 \( 1 - 9.17T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
97 \( 1 + (7.15 - 7.15i)T - 97iT^{2} \)
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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

−10.42812908319031264816802370489, −9.530963892873700430801033454964, −8.895567085062578693586007105153, −7.919680272361375083299389338737, −7.01252487226297621767923968898, −5.86389838964863802217407023288, −5.04739590218116660183105913645, −3.93929380995170875292467909239, −2.52526788756632468889297859855, −1.25033355998033365543095087354, 1.65359283142315068622270411175, 2.76542286429590020942459983312, 3.95073187631193544241613864347, 5.37761794841134496566510645972, 6.15530916914123719290366478333, 6.87223623542750073154687854696, 8.390345478041368997197269060260, 8.877459854298731430198106134120, 9.546171273194241527455457572326, 10.78578930276686650976498312359

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