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Properties

Label 2-1280-40.3-c1-0-33
Degree $2$
Conductor $1280$
Sign $0.640 + 0.767i$
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 + i)3-s + (2 − i)5-s + (−1 − i)7-s − i·9-s − 6·11-s + (1 − i)13-s + (3 + i)15-s + (1 − i)17-s − 4i·19-s − 2i·21-s + (5 − 5i)23-s + (3 − 4i)25-s + (4 − 4i)27-s + 8·29-s + 2i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (0.894 − 0.447i)5-s + (−0.377 − 0.377i)7-s − 0.333i·9-s − 1.80·11-s + (0.277 − 0.277i)13-s + (0.774 + 0.258i)15-s + (0.242 − 0.242i)17-s − 0.917i·19-s − 0.436i·21-s + (1.04 − 1.04i)23-s + (0.600 − 0.800i)25-s + (0.769 − 0.769i)27-s + 1.48·29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.640 + 0.767i$
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: \((2,\ 1280,\ (\ :1/2),\ 0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.996125197\)
\(L(\frac12)\) \(\approx\) \(1.996125197\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-5 + 5i)T - 23iT^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (7 + 7i)T + 47iT^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + (7 - 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (9 + 9i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−9.667528884723615665307041419012, −8.704802103274012488755441816104, −8.286939527238561190360143663378, −7.03314760966549803278375586568, −6.23854823234849410020436460528, −5.10406496873914563676237840731, −4.60246418321684472603823063312, −3.09646210670196029693825747092, −2.62367923068303506980941316916, −0.78009563855881793049807180819, 1.61017715746764157038280964060, 2.59877290287277884750544417019, 3.19298104510118680229330806411, 4.90659092493827824369394162835, 5.66443490782206789889608796529, 6.47192500526470257220498082761, 7.51461260341053648796666301248, 8.010523303179209619967310862233, 8.949366435985731420837697770313, 9.800862196148917985150105075181

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