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Properties

Label 2-1280-40.29-c1-0-30
Degree $2$
Conductor $1280$
Sign $0.948 + 0.316i$
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  + 2·3-s + (2 − i)5-s + 2i·7-s + 9-s − 4i·11-s + 4·13-s + (4 − 2i)15-s − 4i·19-s + 4i·21-s + 2i·23-s + (3 − 4i)25-s − 4·27-s + 2i·29-s − 8i·33-s + (2 + 4i)35-s + ⋯
L(s)  = 1  + 1.15·3-s + (0.894 − 0.447i)5-s + 0.755i·7-s + 0.333·9-s − 1.20i·11-s + 1.10·13-s + (1.03 − 0.516i)15-s − 0.917i·19-s + 0.872i·21-s + 0.417i·23-s + (0.600 − 0.800i)25-s − 0.769·27-s + 0.371i·29-s − 1.39i·33-s + (0.338 + 0.676i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.948 + 0.316i$
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: \((2,\ 1280,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.928696213\)
\(L(\frac12)\) \(\approx\) \(2.928696213\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
good3 \( 1 - 2T + 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−9.115774200879703379344269215017, −8.944329732012852533872773497271, −8.400367180711993083890937552543, −7.33889791434532506640505773025, −5.94665255851497599667597950635, −5.77039376323110168709176252833, −4.37629587990723114153748126098, −3.16500733785694862934791755607, −2.53907267300452578681622799370, −1.26729189908277133068136880264, 1.54513070467557192739459073335, 2.44546607486081798252542501491, 3.51051906362093501264957702027, 4.30501615326470824572477170149, 5.62477918740040340339592610880, 6.52157721483254482640135406635, 7.35893820033940340358296337314, 8.104865261834985748844516581299, 8.961545887259046685843164646129, 9.702448873512547699187233314861

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