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Properties

Label 2-280-280.59-c1-0-39
Degree $2$
Conductor $280$
Sign $0.187 + 0.982i$
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 − 0.743i)2-s + (0.818 − 1.41i)3-s + (0.894 − 1.78i)4-s + (2.22 + 0.187i)5-s + (−0.0691 − 2.31i)6-s + (−2.64 + 0.148i)7-s + (−0.253 − 2.81i)8-s + (0.158 + 0.274i)9-s + (2.82 − 1.43i)10-s + (−2.19 + 3.79i)11-s + (−1.80 − 2.73i)12-s + 2.75i·13-s + (−3.06 + 2.14i)14-s + (2.09 − 3.00i)15-s + (−2.39 − 3.20i)16-s + (0.648 − 1.12i)17-s + ⋯
L(s)  = 1  + (0.850 − 0.525i)2-s + (0.472 − 0.818i)3-s + (0.447 − 0.894i)4-s + (0.996 + 0.0840i)5-s + (−0.0282 − 0.945i)6-s + (−0.998 + 0.0561i)7-s + (−0.0895 − 0.995i)8-s + (0.0529 + 0.0916i)9-s + (0.891 − 0.452i)10-s + (−0.661 + 1.14i)11-s + (−0.520 − 0.789i)12-s + 0.764i·13-s + (−0.819 + 0.572i)14-s + (0.539 − 0.776i)15-s + (−0.599 − 0.800i)16-s + (0.157 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83223 - 1.51592i\)
\(L(\frac12)\) \(\approx\) \(1.83223 - 1.51592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.743i)T \)
5 \( 1 + (-2.22 - 0.187i)T \)
7 \( 1 + (2.64 - 0.148i)T \)
good3 \( 1 + (-0.818 + 1.41i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.19 - 3.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.75iT - 13T^{2} \)
17 \( 1 + (-0.648 + 1.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.86 - 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.136 + 0.236i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.84iT - 29T^{2} \)
31 \( 1 + (2.27 - 3.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.94 + 6.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.95iT - 41T^{2} \)
43 \( 1 + 4.46iT - 43T^{2} \)
47 \( 1 + (-7.22 + 4.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.40 - 9.35i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.31 - 1.91i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.73 - 8.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.32 + 4.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.42iT - 71T^{2} \)
73 \( 1 + (1.84 - 3.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.3 + 7.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (3.15 - 1.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.0199T + 97T^{2} \)
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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

−12.15426440751422609246832025205, −10.57382237392615660149471319485, −9.990282230791470283580077082187, −9.047681157208689775642267890522, −7.37182947087181693080644194985, −6.62680868678907448259024417043, −5.63828290775879218812443082328, −4.29055313412160273455203478978, −2.62251646816365438621156894088, −1.90540485731660223332647984743, 2.81574226457793270901893377038, 3.57193407673272565650061676464, 5.03087899302329513625755038607, 5.95589418271471653110331250406, 6.82924128958343266871342801532, 8.362620117267856834524759869802, 9.116711151125332350173090474968, 10.22743660824293303334222928891, 10.92724550496890567371646561838, 12.61483782018886719164230987463

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