Properties

Label 2-1280-40.27-c1-0-21
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  + (1 − i)3-s + (2 − i)5-s + (−3 + 3i)7-s + i·9-s + 2·11-s + (3 + 3i)13-s + (1 − 3i)15-s + (1 + i)17-s − 4i·19-s + 6i·21-s + (−1 − i)23-s + (3 − 4i)25-s + (4 + 4i)27-s + 10i·31-s + (2 − 2i)33-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (0.894 − 0.447i)5-s + (−1.13 + 1.13i)7-s + 0.333i·9-s + 0.603·11-s + (0.832 + 0.832i)13-s + (0.258 − 0.774i)15-s + (0.242 + 0.242i)17-s − 0.917i·19-s + 1.30i·21-s + (−0.208 − 0.208i)23-s + (0.600 − 0.800i)25-s + (0.769 + 0.769i)27-s + 1.79i·31-s + (0.348 − 0.348i)33-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

\(L(1)\) \(\approx\) \(2.230220880\)
\(L(\frac12)\) \(\approx\) \(2.230220880\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (-3 + 3i)T - 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
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

−9.402456803584422082986961650440, −8.875916381237804046764419913818, −8.442967585645943387558307781780, −7.04691145765800249607407565859, −6.41447180530827700891439005324, −5.71220766864053706978570655988, −4.65789450072451515343506568620, −3.25428908236209704701738066116, −2.38533361247367005137780260175, −1.42990253219613563328555726280, 0.976448526410779716129716993104, 2.65187037517861713573738993739, 3.64492132258854567330998707155, 4.00085779896879103884871738354, 5.70035539540159393762998946890, 6.26544967810012028475264976236, 7.09784449636194623806615124941, 8.054915792416821113508114592346, 9.183588706578470632279672495318, 9.645350837740558908925893226743

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