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Properties

Label 2-1280-20.3-c1-0-0
Degree $2$
Conductor $1280$
Sign $-0.725 - 0.688i$
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  + (−0.456 + 0.456i)3-s + (0.456 − 2.18i)5-s + (−2.79 − 2.79i)7-s + 2.58i·9-s + 4.37i·11-s + (−1.73 − 1.73i)13-s + (0.791 + 1.20i)15-s + (3 − 3i)17-s − 3.46·19-s + 2.55·21-s + (−0.791 + 0.791i)23-s + (−4.58 − 1.99i)25-s + (−2.55 − 2.55i)27-s + 5.29i·29-s + 1.58i·31-s + ⋯
L(s)  = 1  + (−0.263 + 0.263i)3-s + (0.204 − 0.978i)5-s + (−1.05 − 1.05i)7-s + 0.860i·9-s + 1.31i·11-s + (−0.480 − 0.480i)13-s + (0.204 + 0.312i)15-s + (0.727 − 0.727i)17-s − 0.794·19-s + 0.556·21-s + (−0.164 + 0.164i)23-s + (−0.916 − 0.399i)25-s + (−0.490 − 0.490i)27-s + 0.982i·29-s + 0.284i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.725 - 0.688i$
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: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.725 - 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2897457568\)
\(L(\frac12)\) \(\approx\) \(0.2897457568\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.456 + 2.18i)T \)
good3 \( 1 + (0.456 - 0.456i)T - 3iT^{2} \)
7 \( 1 + (2.79 + 2.79i)T + 7iT^{2} \)
11 \( 1 - 4.37iT - 11T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + (0.791 - 0.791i)T - 23iT^{2} \)
29 \( 1 - 5.29iT - 29T^{2} \)
31 \( 1 - 1.58iT - 31T^{2} \)
37 \( 1 + (5.19 - 5.19i)T - 37iT^{2} \)
41 \( 1 - 7.58T + 41T^{2} \)
43 \( 1 + (8.29 - 8.29i)T - 43iT^{2} \)
47 \( 1 + (0.791 + 0.791i)T + 47iT^{2} \)
53 \( 1 + (-2.64 - 2.64i)T + 53iT^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 + 9.66T + 61T^{2} \)
67 \( 1 + (-8.29 - 8.29i)T + 67iT^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (0.582 + 0.582i)T + 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (4.83 - 4.83i)T - 83iT^{2} \)
89 \( 1 + 3.16iT - 89T^{2} \)
97 \( 1 + (-0.582 + 0.582i)T - 97iT^{2} \)
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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

−10.01782784792226277300781732275, −9.453758035001810550962997271402, −8.316461067906695654701752083341, −7.46084365787547442093419276280, −6.82586608539791079977842541737, −5.61733857849532921068173863760, −4.81426393962561805556137229509, −4.20596041067003768068876430160, −2.88900741875531642491218896892, −1.46695915652241174708754803727, 0.12196649610526455167181412303, 2.11995580734854368621407193244, 3.15170224225844111490111025225, 3.81458870462094161053880747338, 5.57864609639983086433439501775, 6.19774566182172589945053719332, 6.51943260324557983592256219938, 7.64000269293688017451236094741, 8.719797247362579738797874042903, 9.347937667069637120058023270102

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