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Properties

Label 2-280-8.5-c1-0-4
Degree $2$
Conductor $280$
Sign $0.886 - 0.462i$
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  + (−0.225 − 1.39i)2-s + 2.07i·3-s + (−1.89 + 0.630i)4-s − i·5-s + (2.90 − 0.469i)6-s + 7-s + (1.30 + 2.50i)8-s − 1.32·9-s + (−1.39 + 0.225i)10-s + 5.25i·11-s + (−1.31 − 3.94i)12-s + 1.61i·13-s + (−0.225 − 1.39i)14-s + 2.07·15-s + (3.20 − 2.39i)16-s + 1.92·17-s + ⋯
L(s)  = 1  + (−0.159 − 0.987i)2-s + 1.20i·3-s + (−0.949 + 0.315i)4-s − 0.447i·5-s + (1.18 − 0.191i)6-s + 0.377·7-s + (0.462 + 0.886i)8-s − 0.441·9-s + (−0.441 + 0.0713i)10-s + 1.58i·11-s + (−0.378 − 1.13i)12-s + 0.448i·13-s + (−0.0603 − 0.373i)14-s + 0.537·15-s + (0.801 − 0.598i)16-s + 0.467·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07054 + 0.262546i\)
\(L(\frac12)\) \(\approx\) \(1.07054 + 0.262546i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.225 + 1.39i)T \)
5 \( 1 + iT \)
7 \( 1 - T \)
good3 \( 1 - 2.07iT - 3T^{2} \)
11 \( 1 - 5.25iT - 11T^{2} \)
13 \( 1 - 1.61iT - 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 - 2.71iT - 19T^{2} \)
23 \( 1 - 3.77T + 23T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 - 4.86T + 31T^{2} \)
37 \( 1 + 7.77iT - 37T^{2} \)
41 \( 1 + 9.39T + 41T^{2} \)
43 \( 1 + 12.9iT - 43T^{2} \)
47 \( 1 + 7.83T + 47T^{2} \)
53 \( 1 + 3.07iT - 53T^{2} \)
59 \( 1 - 9.60iT - 59T^{2} \)
61 \( 1 - 2.70iT - 61T^{2} \)
67 \( 1 + 10.5iT - 67T^{2} \)
71 \( 1 - 0.391T + 71T^{2} \)
73 \( 1 - 2.39T + 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + 4.52iT - 83T^{2} \)
89 \( 1 - 2.27T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−11.92962083238391860634916858216, −10.77140584430880414762370147835, −10.06016561353192487289497008355, −9.423350587046240071389593007373, −8.544959077055330321217112300374, −7.28363791902825071885262202533, −5.20891298072682081758354932392, −4.57304943072323118233476362165, −3.61213438690050070724958350697, −1.84306540626309327728977172790, 1.01012661130198315440273762265, 3.16375332010381895771341459873, 4.98453791631262224051911984820, 6.19066923688276140736057402459, 6.78406588576144333130602957895, 7.974263572389297566429723310441, 8.329876043884270137633738773879, 9.709863061522239037795569174724, 10.89841170691566389082515568164, 11.86191880732386286402038150295

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