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Properties

Label 2-280-280.139-c1-0-7
Degree $2$
Conductor $280$
Sign $-0.927 - 0.372i$
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 + (−1.06 + 2.97i)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 + (−0.337 + 0.941i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.927 - 0.372i$
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: \((2,\ 280,\ (\ :1/2),\ -0.927 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166004 + 0.858445i\)
\(L(\frac12)\) \(\approx\) \(0.166004 + 0.858445i\)
\(L(\frac{3}{2})\) 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

−12.32627529871787140680465542315, −11.34674477306012331657927458940, −10.70930754474287949123636431219, −9.848311149584203356218301322558, −7.930000709815191216846895669725, −6.83060956570469596651204912905, −6.33533687085803411115283656676, −5.44337256480995517596539096763, −4.23156302273776771520025633694, −2.81407155620971436255397767096, 0.57807835353583206995095207600, 2.68464761609195032684606924218, 4.56356398572921258921518879788, 5.26708773758917956356316620289, 6.01289289752950644857060629072, 6.99458475050507248105246336578, 8.853929456826821892637215120102, 9.943200322516702461568529609903, 10.75877764534613469046888978497, 11.60748886265620936960813765962

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