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Properties

Label 2-1280-8.5-c3-0-47
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6i·3-s + 5i·5-s − 34·7-s − 9·9-s + 16i·11-s + 58i·13-s − 30·15-s − 70·17-s − 4i·19-s − 204i·21-s − 134·23-s − 25·25-s + 108i·27-s − 242i·29-s − 100·31-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.447i·5-s − 1.83·7-s − 0.333·9-s + 0.438i·11-s + 1.23i·13-s − 0.516·15-s − 0.998·17-s − 0.0482i·19-s − 2.11i·21-s − 1.21·23-s − 0.200·25-s + 0.769i·27-s − 1.54i·29-s − 0.579·31-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: \(1\)
Selberg data: \((2,\ 1280,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 6iT - 27T^{2} \)
7 \( 1 + 34T + 343T^{2} \)
11 \( 1 - 16iT - 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
17 \( 1 + 70T + 4.91e3T^{2} \)
19 \( 1 + 4iT - 6.85e3T^{2} \)
23 \( 1 + 134T + 1.21e4T^{2} \)
29 \( 1 + 242iT - 2.43e4T^{2} \)
31 \( 1 + 100T + 2.97e4T^{2} \)
37 \( 1 - 438iT - 5.06e4T^{2} \)
41 \( 1 - 138T + 6.89e4T^{2} \)
43 \( 1 - 178iT - 7.95e4T^{2} \)
47 \( 1 + 22T + 1.03e5T^{2} \)
53 \( 1 + 162iT - 1.48e5T^{2} \)
59 \( 1 + 268iT - 2.05e5T^{2} \)
61 \( 1 - 250iT - 2.26e5T^{2} \)
67 \( 1 + 422iT - 3.00e5T^{2} \)
71 \( 1 + 852T + 3.57e5T^{2} \)
73 \( 1 + 306T + 3.89e5T^{2} \)
79 \( 1 - 456T + 4.93e5T^{2} \)
83 \( 1 + 434iT - 5.71e5T^{2} \)
89 \( 1 - 726T + 7.04e5T^{2} \)
97 \( 1 - 1.37e3T + 9.12e5T^{2} \)
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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

−9.587506912426515772653170542610, −8.724008917527551254855206246271, −7.40462246896954233825100931045, −6.51846868918154285693602356426, −6.09372016685551736636052756157, −4.61794446749442080352496139828, −4.03380857909281170993181713376, −3.19845965599820779294171939540, −2.11438454019595056294603807452, 0, 0.72432522782222773782146356481, 2.10398492204946148474757201630, 3.12087215124242464614403202141, 4.04536217251803621567822523093, 5.59919496055218049318046503027, 6.11010763630933158356187288671, 6.98373213895333664042005424253, 7.57706667744623064729707470650, 8.616971778128198817178950251433

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