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Properties

Label 2-1280-40.3-c1-0-8
Degree $2$
Conductor $1280$
Sign $0.973 + 0.229i$
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  + (−1.73 − 1.73i)3-s + (−2 − i)5-s + (1.73 + 1.73i)7-s + 2.99i·9-s − 3.46·11-s + (1 − i)13-s + (1.73 + 5.19i)15-s + (1 − i)17-s + 6.92i·19-s − 5.99i·21-s + (−1.73 + 1.73i)23-s + (3 + 4i)25-s + 4·29-s − 3.46i·31-s + (5.99 + 5.99i)33-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)3-s + (−0.894 − 0.447i)5-s + (0.654 + 0.654i)7-s + 0.999i·9-s − 1.04·11-s + (0.277 − 0.277i)13-s + (0.447 + 1.34i)15-s + (0.242 − 0.242i)17-s + 1.58i·19-s − 1.30i·21-s + (−0.361 + 0.361i)23-s + (0.600 + 0.800i)25-s + 0.742·29-s − 0.622i·31-s + (1.04 + 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8135195114\)
\(L(\frac12)\) \(\approx\) \(0.8135195114\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 1.73i)T + 3iT^{2} \)
7 \( 1 + (-1.73 - 1.73i)T + 7iT^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + (1.73 - 1.73i)T - 23iT^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + (-1.73 - 1.73i)T + 43iT^{2} \)
47 \( 1 + (1.73 + 1.73i)T + 47iT^{2} \)
53 \( 1 + (-7 + 7i)T - 53iT^{2} \)
59 \( 1 - 6.92iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 + (-5.19 + 5.19i)T - 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-12.1 - 12.1i)T + 83iT^{2} \)
89 \( 1 + 8iT - 89T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−9.728085966180043629719815206713, −8.327575378878095820709287966216, −8.059944996275376094095915806533, −7.31372481038622611662331887268, −6.20046281201546766898151929982, −5.49417876278610238580917008599, −4.82787669480671824358480454860, −3.52288164616723216047463528685, −2.05601865727651715919998004771, −0.844412216663648797129275997289, 0.57071229291142128264431326086, 2.67356148997737691957147354396, 3.93090077919749968533425366975, 4.58329646297923248248155734147, 5.22749700962056678580572602532, 6.34242811483932212847056428215, 7.26335686883943130900090418164, 7.992988578973072145542007330374, 8.915206338596683761934500164856, 10.07127256356057522567139878936

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