Properties

Label 2-1280-20.19-c0-0-1
Degree $2$
Conductor $1280$
Sign $-i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − 9-s + 2i·13-s − 25-s + 2i·37-s + 2·41-s i·45-s − 49-s − 2i·53-s − 2·65-s + 81-s + 2·89-s − 2i·117-s + ⋯
L(s)  = 1  + i·5-s − 9-s + 2i·13-s − 25-s + 2i·37-s + 2·41-s i·45-s − 49-s − 2i·53-s − 2·65-s + 81-s + 2·89-s − 2i·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8890633097\)
\(L(\frac12)\) \(\approx\) \(0.8890633097\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.994793330505091035249915327841, −9.316453262918297456476663793718, −8.480920221106173447274895925969, −7.57727930770357627521406680542, −6.59957908161261979085605633843, −6.22696551965542421121709836129, −4.97860869551230558678845950276, −3.94044137499837100849588764277, −2.93826207171072861338336841391, −1.94926018263079764257266318814, 0.75773930440641334800691844760, 2.45287994393897525826934181591, 3.48531571932710923668761114259, 4.64458366134901013714509863897, 5.63658752206409482313247284201, 5.92421089632189173095394856323, 7.54289146383642021304464295970, 8.026880209641628523569417145420, 8.869293515343342907580846639575, 9.482603458082583483086456790282

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