Properties

Label 2-1280-40.27-c1-0-40
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  + (2 − 2i)3-s + (−1 − 2i)5-s + (−2 + 2i)7-s − 5i·9-s + (1 + i)13-s + (−6 − 2i)15-s + (−5 − 5i)17-s − 4i·19-s + 8i·21-s + (−2 − 2i)23-s + (−3 + 4i)25-s + (−4 − 4i)27-s − 4·29-s − 4i·31-s + (6 + 2i)35-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 + (0.277 + 0.277i)13-s + (−1.54 − 0.516i)15-s + (−1.21 − 1.21i)17-s − 0.917i·19-s + 1.74i·21-s + (−0.417 − 0.417i)23-s + (−0.600 + 0.800i)25-s + (−0.769 − 0.769i)27-s − 0.742·29-s − 0.718i·31-s + (1.01 + 0.338i)35-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: $\chi_{1280} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.441969078\)
\(L(\frac12)\) \(\approx\) \(1.441969078\)
\(L(\frac{3}{2})\) 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 \( 1 + 11T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2 + 2i)T + 23iT^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-6 + 6i)T - 43iT^{2} \)
47 \( 1 + (2 - 2i)T - 47iT^{2} \)
53 \( 1 + (7 + 7i)T + 53iT^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + (-10 - 10i)T + 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-3 + 3i)T - 73iT^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + (2 - 2i)T - 83iT^{2} \)
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

−9.126736827397984662897507742928, −8.584297267753593967619216433470, −7.72886987944764369103945800197, −6.97863353916608501655469369985, −6.21933245936249948878567538732, −5.00700533343344961201314165871, −3.88711225932820422659700164371, −2.77130085051307419705626505132, −2.02087816636786727949629038252, −0.48801766974882057292402732214, 2.14672615844771159328365190354, 3.40358533782168554823678821481, 3.68452738103680850149415273790, 4.50204448171704456851054176345, 5.99107627650993733301132069094, 6.83899886389135080126989629111, 7.85023832529147102938108374188, 8.385563852228643003862097865187, 9.413182050107134106519429533585, 9.973679911939139526748356336571

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