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Properties

Label 2-1280-40.29-c1-0-12
Degree $2$
Conductor $1280$
Sign $-0.0613 - 0.998i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
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  + 0.806·3-s + (1.67 + 1.48i)5-s + 2.15i·7-s − 2.35·9-s − 0.387i·11-s − 0.962·13-s + (1.35 + 1.19i)15-s − 1.61i·17-s + 6.31i·19-s + 1.73i·21-s + 6.15i·23-s + (0.612 + 4.96i)25-s − 4.31·27-s − 2i·29-s + 9.92·31-s + ⋯
L(s)  = 1  + 0.465·3-s + (0.749 + 0.662i)5-s + 0.815i·7-s − 0.783·9-s − 0.116i·11-s − 0.266·13-s + (0.348 + 0.308i)15-s − 0.390i·17-s + 1.44i·19-s + 0.379i·21-s + 1.28i·23-s + (0.122 + 0.992i)25-s − 0.829·27-s − 0.371i·29-s + 1.78·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0613 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0613 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.0613 - 0.998i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.0613 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.817449698\)
\(L(\frac12)\) \(\approx\) \(1.817449698\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 1.48i)T \)
good3 \( 1 - 0.806T + 3T^{2} \)
7 \( 1 - 2.15iT - 7T^{2} \)
11 \( 1 + 0.387iT - 11T^{2} \)
13 \( 1 + 0.962T + 13T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 - 6.31iT - 19T^{2} \)
23 \( 1 - 6.15iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 9.92T + 31T^{2} \)
37 \( 1 - 6.57T + 37T^{2} \)
41 \( 1 + 4.57T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 - 4.54iT - 47T^{2} \)
53 \( 1 + 8.96T + 53T^{2} \)
59 \( 1 - 6.31iT - 59T^{2} \)
61 \( 1 - 0.261iT - 61T^{2} \)
67 \( 1 + 9.89T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 - 2.88T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 9.61iT - 97T^{2} \)
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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

−9.758248930196339868661115113089, −9.189463007958550911488723443656, −8.242446377727672034097771900788, −7.61462339889902193714138967173, −6.32835835728099418793445413938, −5.88367327302272606855530722323, −4.95663466285069125706354709223, −3.43395750938636894574526573694, −2.75738206584034249570609290386, −1.77303901588233155962244833102, 0.69442204614492528242168396597, 2.17301472151615180443400491619, 3.10764907866828550411708338943, 4.45228163340157235125809958289, 5.05229496815414267386762718627, 6.24594513251849371471646696574, 6.88553232380991913654159901982, 8.142326371612442182342634542859, 8.556272213829627439869919930977, 9.457935203004123963759187221230

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