Properties

Label 2-1280-8.5-c1-0-3
Degree $2$
Conductor $1280$
Sign $-0.707 + 0.707i$
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  + 2.82i·3-s + i·5-s − 2.82·7-s − 5.00·9-s + 5.65i·11-s + 2i·13-s − 2.82·15-s + 2·17-s − 8.00i·21-s + 2.82·23-s − 25-s − 5.65i·27-s − 6i·29-s − 5.65·31-s − 16.0·33-s + ⋯
L(s)  = 1  + 1.63i·3-s + 0.447i·5-s − 1.06·7-s − 1.66·9-s + 1.70i·11-s + 0.554i·13-s − 0.730·15-s + 0.485·17-s − 1.74i·21-s + 0.589·23-s − 0.200·25-s − 1.08i·27-s − 1.11i·29-s − 1.01·31-s − 2.78·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8668849479\)
\(L(\frac12)\) \(\approx\) \(0.8668849479\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 2.82iT - 3T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 2.82iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.940096655610559686671560551760, −9.581726885645358553047146033818, −9.049696934652772412553276282570, −7.67661100397263353365656214357, −6.86137816883771863988117360982, −5.91061639709618980634821449642, −4.91337796002406823701912059557, −4.12360070308573337447723781820, −3.40064548928288007145308682572, −2.29428942092811512260598712903, 0.37258392630966781462872934426, 1.36067334862514455836791140180, 2.86425756751636377550836488887, 3.49110802942974164431421866743, 5.34084132463577479895481249631, 5.93614132116438903279411387559, 6.73319427278279329214541701374, 7.41782094965223287842466604287, 8.444296096422465025948149731556, 8.753819101325881482354829221044

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