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Properties

Label 2-640-80.27-c1-0-2
Degree $2$
Conductor $640$
Sign $-0.656 - 0.754i$
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  − 2.96·3-s + (0.177 + 2.22i)5-s + (0.115 + 0.115i)7-s + 5.79·9-s + (2.95 − 2.95i)11-s + 1.55i·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 − 8.28·27-s + (−0.289 − 0.289i)29-s + 4.18i·31-s + ⋯
L(s)  = 1  − 1.71·3-s + (0.0793 + 0.996i)5-s + (0.0435 + 0.0435i)7-s + 1.93·9-s + (0.892 − 0.892i)11-s + 0.432i·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 − 1.59·27-s + (−0.0537 − 0.0537i)29-s + 0.751i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.656 - 0.754i$
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: \((2,\ 640,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222940 + 0.489783i\)
\(L(\frac12)\) \(\approx\) \(0.222940 + 0.489783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.177 - 2.22i)T \)
good3 \( 1 + 2.96T + 3T^{2} \)
7 \( 1 + (-0.115 - 0.115i)T + 7iT^{2} \)
11 \( 1 + (-2.95 + 2.95i)T - 11iT^{2} \)
13 \( 1 - 1.55iT - 13T^{2} \)
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 \( 1 + 1.63iT - 37T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 - 6.72iT - 43T^{2} \)
47 \( 1 + (4.38 - 4.38i)T - 47iT^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (-1.63 - 1.63i)T + 59iT^{2} \)
61 \( 1 + (-1.23 + 1.23i)T - 61iT^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 + 8.00T + 71T^{2} \)
73 \( 1 + (1.12 + 1.12i)T + 73iT^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 - 1.62T + 83T^{2} \)
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

−11.12921103737270843569361939570, −10.24678978102023047704507698445, −9.507003396295071514884292713990, −8.100634975152399140493855027239, −6.96796120946460140132166062714, −6.25607357424912688326689520604, −5.79887270533233601092479819235, −4.51264016799954122567983680846, −3.43282217184800335184464160937, −1.53091280724366509216236290937, 0.38244084675756231038762973585, 1.76631109301914768745254464720, 4.15342461566248944136993748374, 4.75435872484354651309461229215, 5.69885424767782112378678742458, 6.45535751007936456637069626668, 7.38620485794177550401257827919, 8.592169418671349670264115917106, 9.621755041595681887519723318353, 10.31561223400869200941975462301

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