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Properties

Label 2-640-80.67-c1-0-17
Degree $2$
Conductor $640$
Sign $-0.852 + 0.522i$
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  − 2.96i·3-s + (2.22 − 0.177i)5-s + (−0.115 − 0.115i)7-s − 5.79·9-s + (−2.95 − 2.95i)11-s − 1.55·13-s + (−0.525 − 6.61i)15-s + (0.299 + 0.299i)17-s + (−2.26 − 2.26i)19-s + (−0.341 + 0.341i)21-s + (4.14 − 4.14i)23-s + (4.93 − 0.790i)25-s + 8.28i·27-s + (0.289 − 0.289i)29-s + 4.18i·31-s + ⋯
L(s)  = 1  − 1.71i·3-s + (0.996 − 0.0793i)5-s + (−0.0435 − 0.0435i)7-s − 1.93·9-s + (−0.892 − 0.892i)11-s − 0.432·13-s + (−0.135 − 1.70i)15-s + (0.0726 + 0.0726i)17-s + (−0.519 − 0.519i)19-s + (−0.0744 + 0.0744i)21-s + (0.864 − 0.864i)23-s + (0.987 − 0.158i)25-s + 1.59i·27-s + (0.0537 − 0.0537i)29-s + 0.751i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.852 + 0.522i$
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: \((2,\ 640,\ (\ :1/2),\ -0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386864 - 1.37259i\)
\(L(\frac12)\) \(\approx\) \(0.386864 - 1.37259i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 0.177i)T \)
good3 \( 1 + 2.96iT - 3T^{2} \)
7 \( 1 + (0.115 + 0.115i)T + 7iT^{2} \)
11 \( 1 + (2.95 + 2.95i)T + 11iT^{2} \)
13 \( 1 + 1.55T + 13T^{2} \)
17 \( 1 + (-0.299 - 0.299i)T + 17iT^{2} \)
19 \( 1 + (2.26 + 2.26i)T + 19iT^{2} \)
23 \( 1 + (-4.14 + 4.14i)T - 23iT^{2} \)
29 \( 1 + (-0.289 + 0.289i)T - 29iT^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 + 1.63T + 37T^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 + (4.38 - 4.38i)T - 47iT^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (-1.63 + 1.63i)T - 59iT^{2} \)
61 \( 1 + (-1.23 - 1.23i)T + 61iT^{2} \)
67 \( 1 + 2.49T + 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (-1.12 - 1.12i)T + 73iT^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 - 1.62iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + (-9.69 - 9.69i)T + 97iT^{2} \)
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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

−10.41457526925671812798686306948, −9.050118952436220798321804218040, −8.433534972965240261746148565738, −7.43620398887230990943916484381, −6.65690720470573014663205907423, −5.90480915928441851354273243105, −5.02434052858071085535213852384, −2.92503882896414934774289907753, −2.14526164299037979015195225452, −0.75370939085634236552201166155, 2.26138598187087367362768344333, 3.37666752676865279711435769727, 4.66166327608639006782492325143, 5.20255532962984912417526742684, 6.14533841729443917449116002079, 7.47791553759515892139311292826, 8.663072792296976728760111796019, 9.604580762766658485567047858477, 9.904805495968235054454259191965, 10.63935201292867512142708076531

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