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Properties

Label 2-280-280.139-c1-0-19
Degree $2$
Conductor $280$
Sign $0.699 - 0.714i$
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  + (−1.20 + 0.741i)2-s + 2.57·3-s + (0.901 − 1.78i)4-s + (0.460 + 2.18i)5-s + (−3.10 + 1.91i)6-s + (2.31 − 1.28i)7-s + (0.236 + 2.81i)8-s + 3.65·9-s + (−2.17 − 2.29i)10-s − 3.59·11-s + (2.32 − 4.60i)12-s + 1.33i·13-s + (−1.83 + 3.25i)14-s + (1.18 + 5.64i)15-s + (−2.37 − 3.21i)16-s + 2.88·17-s + ⋯
L(s)  = 1  + (−0.851 + 0.523i)2-s + 1.48·3-s + (0.450 − 0.892i)4-s + (0.205 + 0.978i)5-s + (−1.26 + 0.780i)6-s + (0.874 − 0.484i)7-s + (0.0837 + 0.996i)8-s + 1.21·9-s + (−0.688 − 0.725i)10-s − 1.08·11-s + (0.671 − 1.32i)12-s + 0.370i·13-s + (−0.491 + 0.870i)14-s + (0.306 + 1.45i)15-s + (−0.593 − 0.804i)16-s + 0.699·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.699 - 0.714i$
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: \((2,\ 280,\ (\ :1/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35035 + 0.568162i\)
\(L(\frac12)\) \(\approx\) \(1.35035 + 0.568162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.741i)T \)
5 \( 1 + (-0.460 - 2.18i)T \)
7 \( 1 + (-2.31 + 1.28i)T \)
good3 \( 1 - 2.57T + 3T^{2} \)
11 \( 1 + 3.59T + 11T^{2} \)
13 \( 1 - 1.33iT - 13T^{2} \)
17 \( 1 - 2.88T + 17T^{2} \)
19 \( 1 + 5.38iT - 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 - 1.88iT - 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + 4.64T + 37T^{2} \)
41 \( 1 - 0.606iT - 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + 6.92iT - 47T^{2} \)
53 \( 1 + 6.31T + 53T^{2} \)
59 \( 1 + 1.39iT - 59T^{2} \)
61 \( 1 + 3.80T + 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 + 8.19iT - 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + 7.04iT - 89T^{2} \)
97 \( 1 - 6.26T + 97T^{2} \)
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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

−11.63616338433775591577106211270, −10.62496545619721985491575624751, −9.984204440602276981998671317267, −8.987103172541973125725032186757, −7.896712843129824294352959786100, −7.63347678376900513559267520253, −6.43719301708771755755647533908, −4.84871208388352135483460839119, −3.05802410841799355444181577842, −1.98296520033794375883853575239, 1.65385902080425449313669594657, 2.71248255774427922541156976708, 4.09165101304245989298626315341, 5.59707832251810674705228630844, 7.69745938348357811500608724426, 8.146759257269620547344531404272, 8.676533108161101427139903362225, 9.750320306320228540268726576381, 10.39049004366456001382410800422, 11.90202296412758774439114270717

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