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Properties

Label 2-1280-8.5-c3-0-76
Degree $2$
Conductor $1280$
Sign $0.707 + 0.707i$
Analytic cond. $75.5224$
Root an. cond. $8.69036$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.898i·3-s − 5i·5-s + 28.4·7-s + 26.1·9-s + 2.60i·11-s − 33.1i·13-s + 4.49·15-s + 119.·17-s − 143. i·19-s + 25.6i·21-s + 113.·23-s − 25·25-s + 47.8i·27-s − 67.6i·29-s − 117.·31-s + ⋯
L(s)  = 1  + 0.173i·3-s − 0.447i·5-s + 1.53·7-s + 0.970·9-s + 0.0714i·11-s − 0.708i·13-s + 0.0773·15-s + 1.70·17-s − 1.72i·19-s + 0.266i·21-s + 1.03·23-s − 0.200·25-s + 0.340i·27-s − 0.432i·29-s − 0.680·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(75.5224\)
Root analytic conductor: \(8.69036\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.163266032\)
\(L(\frac12)\) \(\approx\) \(3.163266032\)
\(L(\frac{5}{2})\) 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 \( 1 + 5iT \)
good3 \( 1 - 0.898iT - 27T^{2} \)
7 \( 1 - 28.4T + 343T^{2} \)
11 \( 1 - 2.60iT - 1.33e3T^{2} \)
13 \( 1 + 33.1iT - 2.19e3T^{2} \)
17 \( 1 - 119.T + 4.91e3T^{2} \)
19 \( 1 + 143. iT - 6.85e3T^{2} \)
23 \( 1 - 113.T + 1.21e4T^{2} \)
29 \( 1 + 67.6iT - 2.43e4T^{2} \)
31 \( 1 + 117.T + 2.97e4T^{2} \)
37 \( 1 - 256. iT - 5.06e4T^{2} \)
41 \( 1 + 245.T + 6.89e4T^{2} \)
43 \( 1 + 369. iT - 7.95e4T^{2} \)
47 \( 1 - 76.3T + 1.03e5T^{2} \)
53 \( 1 + 40.4iT - 1.48e5T^{2} \)
59 \( 1 - 457. iT - 2.05e5T^{2} \)
61 \( 1 + 477. iT - 2.26e5T^{2} \)
67 \( 1 - 602. iT - 3.00e5T^{2} \)
71 \( 1 + 1.09e3T + 3.57e5T^{2} \)
73 \( 1 + 117.T + 3.89e5T^{2} \)
79 \( 1 + 858.T + 4.93e5T^{2} \)
83 \( 1 - 565. iT - 5.71e5T^{2} \)
89 \( 1 - 625.T + 7.04e5T^{2} \)
97 \( 1 - 805.T + 9.12e5T^{2} \)
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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

−9.124961670940072941610513365666, −8.369267045393644112786841903595, −7.58889249757990825720485848534, −6.98501242447774941671329054073, −5.45944520622883554537138411308, −5.02312105109601906877774180563, −4.21321380949870871165837105262, −2.98337190807375428236840611508, −1.60809677083233058143183313472, −0.835194105650410850989970832789, 1.29071222007278586827582284794, 1.78736266602424223344411235548, 3.32491939615143216100974882552, 4.25900416962998201938004156683, 5.17322489537262865907186091157, 6.01952867849120048375254767994, 7.23314080638599040838426951596, 7.63616088857887225868396573326, 8.417957835715909740453223701107, 9.467991763518713495519774511310

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