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Properties

Label 2-640-80.67-c1-0-19
Degree $2$
Conductor $640$
Sign $-0.880 - 0.473i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.179930 + 0.713979i\)
\(L(\frac12)\) \(\approx\) \(0.179930 + 0.713979i\)
\(L(\frac{3}{2})\) 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

−9.993904452351200224556033284900, −8.891240663426026533120321046847, −8.125174654082322187398124303294, −7.45641946237273144692937792276, −6.76372046078297927705197408709, −5.54759303630207946035600778549, −4.65624353036751989305736424652, −2.99002118778259211697633113358, −1.77206318595329899296613565945, −0.38784597263030990389281731674, 2.68943770901227998069785373143, 3.60839464740730455726339199995, 4.59298487646303949386196022217, 5.23192172535350987735148907002, 6.70550517265486631400709705869, 7.56892008967107727553239274287, 8.688194752601963526616780755626, 9.503914574161298581057408511615, 10.32262840674642007638825502664, 10.85116032131889130506448586914

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