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Properties

Label 2-280-280.139-c1-0-26
Degree $2$
Conductor $280$
Sign $0.270 + 0.962i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−0.918 + 1.07i)2-s − 1.19·3-s + (−0.313 − 1.97i)4-s + (2.22 + 0.234i)5-s + (1.09 − 1.28i)6-s + (−2.11 − 1.58i)7-s + (2.41 + 1.47i)8-s − 1.58·9-s + (−2.29 + 2.17i)10-s − 3.65·11-s + (0.373 + 2.35i)12-s − 5.56i·13-s + (3.65 − 0.822i)14-s + (−2.64 − 0.279i)15-s + (−3.80 + 1.23i)16-s + 0.808·17-s + ⋯
L(s)  = 1  + (−0.649 + 0.760i)2-s − 0.687·3-s + (−0.156 − 0.987i)4-s + (0.994 + 0.105i)5-s + (0.446 − 0.522i)6-s + (−0.800 − 0.599i)7-s + (0.852 + 0.521i)8-s − 0.527·9-s + (−0.725 + 0.688i)10-s − 1.10·11-s + (0.107 + 0.678i)12-s − 1.54i·13-s + (0.975 − 0.219i)14-s + (−0.683 − 0.0722i)15-s + (−0.950 + 0.309i)16-s + 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373479 - 0.283054i\)
\(L(\frac12)\) \(\approx\) \(0.373479 - 0.283054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.918 - 1.07i)T \)
5 \( 1 + (-2.22 - 0.234i)T \)
7 \( 1 + (2.11 + 1.58i)T \)
good3 \( 1 + 1.19T + 3T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 + 5.56iT - 13T^{2} \)
17 \( 1 - 0.808T + 17T^{2} \)
19 \( 1 + 4.54iT - 19T^{2} \)
23 \( 1 - 1.75T + 23T^{2} \)
29 \( 1 + 8.36iT - 29T^{2} \)
31 \( 1 + 4.73T + 31T^{2} \)
37 \( 1 + 6.15T + 37T^{2} \)
41 \( 1 - 7.65iT - 41T^{2} \)
43 \( 1 + 2.66iT - 43T^{2} \)
47 \( 1 + 4.79iT - 47T^{2} \)
53 \( 1 + 2.98T + 53T^{2} \)
59 \( 1 + 8.92iT - 59T^{2} \)
61 \( 1 - 4.08T + 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 - 8.17iT - 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 2.49iT - 79T^{2} \)
83 \( 1 - 5.75T + 83T^{2} \)
89 \( 1 + 4.95iT - 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−11.25835898651014441054613630906, −10.38327073563453384684820983495, −9.992198284322751612229693438721, −8.798288552272701486504778560446, −7.67417482986994518551211694803, −6.62952236997293920796169767397, −5.72272749487224354902590291119, −5.11223758708300842961236513997, −2.78487683493534377628811868507, −0.45256969043558577848573247443, 1.92450556472681959441494390945, 3.20180678719793615573798593134, 5.03369603558841438016221143563, 6.04341805073640506033174484018, 7.13360926147933164848473220215, 8.697311869190927898467502967069, 9.258214029446342925536882648416, 10.28937655422185420060381886333, 10.92883287579702236825714369475, 12.09336476678506118374260366929

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