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Properties

Label 2-1280-20.7-c1-0-35
Degree $2$
Conductor $1280$
Sign $-0.169 + 0.985i$
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  + (−0.569 − 0.569i)3-s + (1.48 − 1.67i)5-s + (2.93 − 2.93i)7-s − 2.35i·9-s + 3.04i·11-s + (1.80 − 1.80i)13-s + (−1.79 + 0.110i)15-s + (−3.96 − 3.96i)17-s + 7.56·19-s − 3.35·21-s + (−1.25 − 1.25i)23-s + (−0.612 − 4.96i)25-s + (−3.04 + 3.04i)27-s + 3.61i·29-s + 8.15i·31-s + ⋯
L(s)  = 1  + (−0.329 − 0.329i)3-s + (0.662 − 0.749i)5-s + (1.11 − 1.11i)7-s − 0.783i·9-s + 0.919i·11-s + (0.500 − 0.500i)13-s + (−0.464 + 0.0285i)15-s + (−0.961 − 0.961i)17-s + 1.73·19-s − 0.731·21-s + (−0.260 − 0.260i)23-s + (−0.122 − 0.992i)25-s + (−0.586 + 0.586i)27-s + 0.670i·29-s + 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.889538159\)
\(L(\frac12)\) \(\approx\) \(1.889538159\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 1.67i)T \)
good3 \( 1 + (0.569 + 0.569i)T + 3iT^{2} \)
7 \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \)
11 \( 1 - 3.04iT - 11T^{2} \)
13 \( 1 + (-1.80 + 1.80i)T - 13iT^{2} \)
17 \( 1 + (3.96 + 3.96i)T + 17iT^{2} \)
19 \( 1 - 7.56T + 19T^{2} \)
23 \( 1 + (1.25 + 1.25i)T + 23iT^{2} \)
29 \( 1 - 3.61iT - 29T^{2} \)
31 \( 1 - 8.15iT - 31T^{2} \)
37 \( 1 + (-3.54 - 3.54i)T + 37iT^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 + (-0.569 - 0.569i)T + 43iT^{2} \)
47 \( 1 + (4.62 - 4.62i)T - 47iT^{2} \)
53 \( 1 + (0.768 - 0.768i)T - 53iT^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 + 6.96T + 61T^{2} \)
67 \( 1 + (6.22 - 6.22i)T - 67iT^{2} \)
71 \( 1 + 6.97iT - 71T^{2} \)
73 \( 1 + (-9.31 + 9.31i)T - 73iT^{2} \)
79 \( 1 - 9.03T + 79T^{2} \)
83 \( 1 + (6.99 + 6.99i)T + 83iT^{2} \)
89 \( 1 + 6.70iT - 89T^{2} \)
97 \( 1 + (-1.38 - 1.38i)T + 97iT^{2} \)
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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

−9.452771204948791560140831051853, −8.704479099664999335603682438237, −7.66166275674255819247733145413, −7.06994516066616263804110398024, −6.14291291956886355755620679214, −4.97587440420259769464433883417, −4.65041792779773043518273351190, −3.27286788673638626133535809311, −1.63548937841939678951322133896, −0.901940178534577478159393475097, 1.73992190754344932018226859435, 2.53613651041171108591309489901, 3.85623579935000409351135020655, 5.03780623782846579412787505508, 5.73499244825735760746380529199, 6.26494330789414457984612301189, 7.59981187775999926545433380470, 8.270501607865440004163271884575, 9.130721323872221988850343780329, 9.901258203610409390279141045759

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