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Properties

Label 2-640-80.27-c1-0-8
Degree $2$
Conductor $640$
Sign $0.459 + 0.888i$
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  − 1.96·3-s + (1.42 + 1.72i)5-s + (−1.60 − 1.60i)7-s + 0.851·9-s + (−0.754 + 0.754i)11-s − 5.94i·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 + 4.21·27-s + (1.44 + 1.44i)29-s − 3.60i·31-s + ⋯
L(s)  = 1  − 1.13·3-s + (0.635 + 0.771i)5-s + (−0.605 − 0.605i)7-s + 0.283·9-s + (−0.227 + 0.227i)11-s − 1.64i·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 + 0.811·27-s + (0.268 + 0.268i)29-s − 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.459 + 0.888i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.459 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714664 - 0.434747i\)
\(L(\frac12)\) \(\approx\) \(0.714664 - 0.434747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 1.72i)T \)
good3 \( 1 + 1.96T + 3T^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + (0.754 - 0.754i)T - 11iT^{2} \)
13 \( 1 + 5.94iT - 13T^{2} \)
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 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 6.93iT - 41T^{2} \)
43 \( 1 + 9.91iT - 43T^{2} \)
47 \( 1 + (-0.104 + 0.104i)T - 47iT^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (-3.46 - 3.46i)T + 59iT^{2} \)
61 \( 1 + (0.680 - 0.680i)T - 61iT^{2} \)
67 \( 1 - 9.04iT - 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
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

−10.56940646152893406149825660582, −10.03028826642704406697236722680, −8.745169527879469769118723927817, −7.46949094871565129364366878995, −6.74834318663532971400499672601, −5.80542007198480114440543339738, −5.30494431994323182113025756205, −3.74254073792445409241940356003, −2.59520348301399255254067052486, −0.57637253788952953617929154322, 1.30427814992324450608311027261, 2.90572756129823481318467412345, 4.62351894349144265346216401457, 5.26771279401203518559372684343, 6.20965329757236154499428202537, 6.75583180230037392583380215643, 8.248266375292200787520772934241, 9.274442101309305161657579954026, 9.662150862019926373306939235429, 10.86460098389738780772663811244

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