Properties

Label 2-640-80.67-c1-0-8
Degree $2$
Conductor $640$
Sign $0.906 - 0.423i$
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  + 1.96i·3-s + (1.72 − 1.42i)5-s + (−1.60 − 1.60i)7-s − 0.851·9-s + (−0.754 − 0.754i)11-s + 5.94·13-s + (2.79 + 3.38i)15-s + (1.95 + 1.95i)17-s + (0.780 + 0.780i)19-s + (3.14 − 3.14i)21-s + (4.93 − 4.93i)23-s + (0.956 − 4.90i)25-s + 4.21i·27-s + (−1.44 + 1.44i)29-s + 3.60i·31-s + ⋯
L(s)  = 1  + 1.13i·3-s + (0.771 − 0.635i)5-s + (−0.605 − 0.605i)7-s − 0.283·9-s + (−0.227 − 0.227i)11-s + 1.64·13-s + (0.720 + 0.874i)15-s + (0.474 + 0.474i)17-s + (0.179 + 0.179i)19-s + (0.686 − 0.686i)21-s + (1.02 − 1.02i)23-s + (0.191 − 0.981i)25-s + 0.811i·27-s + (−0.268 + 0.268i)29-s + 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.906 - 0.423i$
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: \((2,\ 640,\ (\ :1/2),\ 0.906 - 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70256 + 0.377895i\)
\(L(\frac12)\) \(\approx\) \(1.70256 + 0.377895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 1.42i)T \)
good3 \( 1 - 1.96iT - 3T^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + (0.754 + 0.754i)T + 11iT^{2} \)
13 \( 1 - 5.94T + 13T^{2} \)
17 \( 1 + (-1.95 - 1.95i)T + 17iT^{2} \)
19 \( 1 + (-0.780 - 0.780i)T + 19iT^{2} \)
23 \( 1 + (-4.93 + 4.93i)T - 23iT^{2} \)
29 \( 1 + (1.44 - 1.44i)T - 29iT^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 - 9.91T + 43T^{2} \)
47 \( 1 + (0.104 - 0.104i)T - 47iT^{2} \)
53 \( 1 + 4.03iT - 53T^{2} \)
59 \( 1 + (3.46 - 3.46i)T - 59iT^{2} \)
61 \( 1 + (0.680 + 0.680i)T + 61iT^{2} \)
67 \( 1 - 9.04T + 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 4.23iT - 83T^{2} \)
89 \( 1 + 0.0426T + 89T^{2} \)
97 \( 1 + (1.91 + 1.91i)T + 97iT^{2} \)
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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

−10.51386092292460982245528869016, −9.877040038554430096132853065861, −8.976151996912927444835233416662, −8.419754670649462272538367714968, −6.91678804439733614122713322338, −5.96224349924787652404987031546, −5.07840204399338247672024505629, −4.05139448560752814189224981447, −3.20923524124951771655446620228, −1.25537026815968796956534729062, 1.33357409306769378851166349020, 2.50506620680367029865830279577, 3.56749660924975104498012069808, 5.46092868495752374659149027135, 6.10222754823957604586925932690, 6.93462949778557543515832064835, 7.62795864056894210656090261534, 8.860299219482513212165313095310, 9.533398876666458868453402468425, 10.58856533482318949822207855964

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