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Properties

Label 2-640-80.67-c1-0-16
Degree $2$
Conductor $640$
Sign $-0.336 + 0.941i$
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  + 0.496i·3-s + (0.987 − 2.00i)5-s + (−1.55 − 1.55i)7-s + 2.75·9-s + (−4.19 − 4.19i)11-s − 5.09·13-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 + 2.85i·27-s + (2.24 − 2.24i)29-s + 0.818i·31-s + ⋯
L(s)  = 1  + 0.286i·3-s + (0.441 − 0.897i)5-s + (−0.587 − 0.587i)7-s + 0.917·9-s + (−1.26 − 1.26i)11-s − 1.41·13-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 + 0.549i·27-s + (0.417 − 0.417i)29-s + 0.146i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.336 + 0.941i$
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: \((2,\ 640,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620248 - 0.880235i\)
\(L(\frac12)\) \(\approx\) \(0.620248 - 0.880235i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.987 + 2.00i)T \)
good3 \( 1 - 0.496iT - 3T^{2} \)
7 \( 1 + (1.55 + 1.55i)T + 7iT^{2} \)
11 \( 1 + (4.19 + 4.19i)T + 11iT^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
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 \( 1 - 5.12T + 37T^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 + 4.49T + 43T^{2} \)
47 \( 1 + (-4.29 + 4.29i)T - 47iT^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 + (-7.65 + 7.65i)T - 59iT^{2} \)
61 \( 1 + (-1.90 - 1.90i)T + 61iT^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (2.70 + 2.70i)T + 73iT^{2} \)
79 \( 1 + 8.32T + 79T^{2} \)
83 \( 1 - 9.17iT - 83T^{2} \)
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

−9.958802951992994071687143348892, −9.827051095582669463798511130894, −8.572694372300185780362881265609, −7.75077683169949128464555286582, −6.76252456619675331383960690724, −5.55667190009179261444477307454, −4.84003420850240554774100768778, −3.74122626386133357224659122519, −2.36256212886308956952516405679, −0.55107478551983923394649089923, 2.13532948313485681716394508686, 2.77332533791153255317734225040, 4.41848949525901923605801643796, 5.42150451897480161682667528145, 6.56375809321738516419604755781, 7.24065691484951834528895103938, 7.929763419572947096130316819201, 9.489976305115520714790533115837, 9.976072996943968916067459539283, 10.51273347714090635056889627202

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