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Properties

Label 2-280-280.69-c2-0-2
Degree $2$
Conductor $280$
Sign $-0.993 - 0.116i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.94 − 0.448i)2-s + 3.68i·3-s + (3.59 + 1.75i)4-s + (0.693 − 4.95i)5-s + (1.65 − 7.18i)6-s + (−4.51 − 5.34i)7-s + (−6.22 − 5.02i)8-s − 4.60·9-s + (−3.57 + 9.33i)10-s + 10.2i·11-s + (−6.45 + 13.2i)12-s + 14.4i·13-s + (6.40 + 12.4i)14-s + (18.2 + 2.55i)15-s + (9.87 + 12.5i)16-s − 16.4·17-s + ⋯
L(s)  = 1  + (−0.974 − 0.224i)2-s + 1.22i·3-s + (0.899 + 0.437i)4-s + (0.138 − 0.990i)5-s + (0.276 − 1.19i)6-s + (−0.645 − 0.763i)7-s + (−0.778 − 0.628i)8-s − 0.512·9-s + (−0.357 + 0.933i)10-s + 0.928i·11-s + (−0.538 + 1.10i)12-s + 1.11i·13-s + (0.457 + 0.889i)14-s + (1.21 + 0.170i)15-s + (0.617 + 0.786i)16-s − 0.967·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ -0.993 - 0.116i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0147164 + 0.251295i\)
\(L(\frac12)\) \(\approx\) \(0.0147164 + 0.251295i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 + 0.448i)T \)
5 \( 1 + (-0.693 + 4.95i)T \)
7 \( 1 + (4.51 + 5.34i)T \)
good3 \( 1 - 3.68iT - 9T^{2} \)
11 \( 1 - 10.2iT - 121T^{2} \)
13 \( 1 - 14.4iT - 169T^{2} \)
17 \( 1 + 16.4T + 289T^{2} \)
19 \( 1 + 31.0T + 361T^{2} \)
23 \( 1 - 20.2iT - 529T^{2} \)
29 \( 1 + 56.8iT - 841T^{2} \)
31 \( 1 - 25.7iT - 961T^{2} \)
37 \( 1 + 66.0T + 1.36e3T^{2} \)
41 \( 1 + 0.504iT - 1.68e3T^{2} \)
43 \( 1 - 26.2T + 1.84e3T^{2} \)
47 \( 1 - 20.2T + 2.20e3T^{2} \)
53 \( 1 - 53.0T + 2.80e3T^{2} \)
59 \( 1 + 59.3T + 3.48e3T^{2} \)
61 \( 1 + 69.5T + 3.72e3T^{2} \)
67 \( 1 + 61.9T + 4.48e3T^{2} \)
71 \( 1 + 25.5T + 5.04e3T^{2} \)
73 \( 1 - 34.7T + 5.32e3T^{2} \)
79 \( 1 - 64.4T + 6.24e3T^{2} \)
83 \( 1 + 9.27iT - 6.88e3T^{2} \)
89 \( 1 + 22.0iT - 7.92e3T^{2} \)
97 \( 1 + 74.3T + 9.40e3T^{2} \)
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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

−11.88222708643767419426981106089, −10.73860656560442407390155187808, −10.08035913170088753038833770984, −9.297277922367661515111300443214, −8.782211314618412352991729091250, −7.36620261091583095004281402885, −6.31069539954684018542845269843, −4.55081564600453566070234592616, −3.92094566234896972194267928058, −1.92329382153349471305551769172, 0.15826951532881535994004879511, 2.07518791419067587272807740350, 3.02476011343835492134203686320, 5.81626624839110401593201789029, 6.46411893446349825014852433110, 7.15461589069133942874027940405, 8.307930208429174183717180896303, 8.967414797830512082096584093735, 10.46505759430840033299374492489, 10.86770201074609701423812931644

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