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Properties

Label 2-1280-20.3-c1-0-36
Degree $2$
Conductor $1280$
Sign $-0.850 + 0.525i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)3-s + (1 − 2i)5-s + (2 + 2i)7-s − 5i·9-s − 4i·11-s + (−3 − 3i)13-s + (2 + 6i)15-s + (−3 + 3i)17-s − 8·21-s + (−6 + 6i)23-s + (−3 − 4i)25-s + (4 + 4i)27-s + 2i·29-s + 4i·31-s + (8 + 8i)33-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s + (0.447 − 0.894i)5-s + (0.755 + 0.755i)7-s − 1.66i·9-s − 1.20i·11-s + (−0.832 − 0.832i)13-s + (0.516 + 1.54i)15-s + (−0.727 + 0.727i)17-s − 1.74·21-s + (−1.25 + 1.25i)23-s + (−0.600 − 0.800i)25-s + (0.769 + 0.769i)27-s + 0.371i·29-s + 0.718i·31-s + (1.39 + 1.39i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good3 \( 1 + (2 - 2i)T - 3iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (6 - 6i)T - 23iT^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 + (6 + 6i)T + 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-6 - 6i)T + 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (6 - 6i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-11 + 11i)T - 97iT^{2} \)
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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

−9.510068305118607769667969282961, −8.581502404159138738954688101602, −8.060088137276290538705919340885, −6.40657561447329141454833232388, −5.61326405579647884905062035838, −5.24383173031467372298135042947, −4.49158944607474995317860323441, −3.33950870353528692256487602720, −1.68375498005259898135911888355, 0, 1.74207533632421601016936417025, 2.33522688916145357185460140694, 4.30475816928882914416401626860, 4.96278191408211701351926304571, 6.14587873090926796891878724028, 6.78014348839458342586090814749, 7.30227835306332592970253743319, 7.908996560457538008532018614127, 9.408887756577383664907424207566

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