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Properties

Label 2-280-280.107-c1-0-11
Degree $2$
Conductor $280$
Sign $-0.670 - 0.741i$
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  + (0.435 + 1.34i)2-s + (1.80 + 0.482i)3-s + (−1.62 + 1.17i)4-s + (−2.22 + 0.227i)5-s + (0.135 + 2.63i)6-s + (0.327 + 2.62i)7-s + (−2.28 − 1.66i)8-s + (0.416 + 0.240i)9-s + (−1.27 − 2.89i)10-s + (1.74 + 3.02i)11-s + (−3.48 + 1.33i)12-s + (0.910 + 0.910i)13-s + (−3.38 + 1.58i)14-s + (−4.11 − 0.665i)15-s + (1.25 − 3.79i)16-s + (0.637 − 2.37i)17-s + ⋯
L(s)  = 1  + (0.308 + 0.951i)2-s + (1.04 + 0.278i)3-s + (−0.810 + 0.586i)4-s + (−0.994 + 0.101i)5-s + (0.0552 + 1.07i)6-s + (0.123 + 0.992i)7-s + (−0.807 − 0.590i)8-s + (0.138 + 0.0802i)9-s + (−0.403 − 0.915i)10-s + (0.525 + 0.910i)11-s + (−1.00 + 0.384i)12-s + (0.252 + 0.252i)13-s + (−0.905 + 0.423i)14-s + (−1.06 − 0.171i)15-s + (0.312 − 0.949i)16-s + (0.154 − 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.670 - 0.741i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.670 - 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.642913 + 1.44749i\)
\(L(\frac12)\) \(\approx\) \(0.642913 + 1.44749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.435 - 1.34i)T \)
5 \( 1 + (2.22 - 0.227i)T \)
7 \( 1 + (-0.327 - 2.62i)T \)
good3 \( 1 + (-1.80 - 0.482i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.74 - 3.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.910 - 0.910i)T + 13iT^{2} \)
17 \( 1 + (-0.637 + 2.37i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.95 - 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.02 + 0.275i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 + (-7.85 + 4.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.893 + 3.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.98T + 41T^{2} \)
43 \( 1 + (3.03 - 3.03i)T - 43iT^{2} \)
47 \( 1 + (2.85 + 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.17 - 11.8i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.06 + 1.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.17 - 1.83i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.12 - 11.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.53iT - 71T^{2} \)
73 \( 1 + (-10.7 - 2.88i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.88 + 4.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.03 + 9.03i)T - 83iT^{2} \)
89 \( 1 + (13.2 + 7.64i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.94 - 7.94i)T + 97iT^{2} \)
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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

−12.07792049804844580732099269860, −11.78580772151912748967618010098, −9.840463371467720000565602849322, −9.025074604954904441333205456086, −8.312469917782399812141753558396, −7.51424857186304932493811579161, −6.40082833558027785753341763649, −4.97263036479204138657490031233, −3.92429164928190524943001682731, −2.84776772324604461060885298077, 1.11765513242050981272277197328, 3.09387070789955628695510266753, 3.69522765249828582710220315949, 4.93497613047299283069464768088, 6.67293296482110432877940818242, 8.118997864291971864527202999748, 8.451974535465057186245943077362, 9.674841917991102010867525900934, 10.82116993444162944126453468307, 11.44835384109545919637513088894

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