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Properties

Label 2-1280-5.4-c1-0-27
Degree $2$
Conductor $1280$
Sign $0.774 + 0.632i$
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  + 2.44i·3-s + (−1.73 − 1.41i)5-s + 1.41i·7-s − 2.99·9-s + 2·11-s − 5.65i·13-s + (3.46 − 4.24i)15-s − 4.89i·17-s − 6·19-s − 3.46·21-s − 7.07i·23-s + (0.999 + 4.89i)25-s − 6.92·29-s + 6.92·31-s + 4.89i·33-s + ⋯
L(s)  = 1  + 1.41i·3-s + (−0.774 − 0.632i)5-s + 0.534i·7-s − 0.999·9-s + 0.603·11-s − 1.56i·13-s + (0.894 − 1.09i)15-s − 1.18i·17-s − 1.37·19-s − 0.755·21-s − 1.47i·23-s + (0.199 + 0.979i)25-s − 1.28·29-s + 1.24·31-s + 0.852i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.003823334\)
\(L(\frac12)\) \(\approx\) \(1.003823334\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 1.41i)T \)
good3 \( 1 - 2.44iT - 3T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 7.07iT - 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 + 4.24iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 - 3.46T + 61T^{2} \)
67 \( 1 + 2.44iT - 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 6.92T + 79T^{2} \)
83 \( 1 + 12.2iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−9.532973672675416218088916761604, −8.796147878616663161993158683328, −8.315813668906354383076012740425, −7.23968065063946833273507693418, −5.97448181530948798255161037722, −5.10984353791789850908754551248, −4.43932208403193096701780525197, −3.66288589633197658939459625567, −2.60892764683498982900389391637, −0.44789852922513516232191787121, 1.33548764282829712681934600340, 2.25742738010312611172239463610, 3.74567981292682963658515094282, 4.32572139969945155905295544977, 6.09794092419407075371583347285, 6.59307601125213246059368864542, 7.24547358200301256571612831751, 7.920191909012525070461414269684, 8.689993307304261888317145720145, 9.716100369216976060982875227738

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