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Properties

Label 2-280-56.37-c1-0-19
Degree $2$
Conductor $280$
Sign $0.399 + 0.916i$
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  + (1.14 − 0.832i)2-s + (−2.56 + 1.48i)3-s + (0.614 − 1.90i)4-s + (−0.866 − 0.5i)5-s + (−1.69 + 3.82i)6-s + (2.64 − 0.0805i)7-s + (−0.882 − 2.68i)8-s + (2.88 − 4.99i)9-s + (−1.40 + 0.149i)10-s + (3.88 − 2.24i)11-s + (1.24 + 5.78i)12-s − 3.72i·13-s + (2.95 − 2.29i)14-s + 2.96·15-s + (−3.24 − 2.33i)16-s + (−1.47 − 2.56i)17-s + ⋯
L(s)  = 1  + (0.808 − 0.588i)2-s + (−1.48 + 0.854i)3-s + (0.307 − 0.951i)4-s + (−0.387 − 0.223i)5-s + (−0.693 + 1.56i)6-s + (0.999 − 0.0304i)7-s + (−0.311 − 0.950i)8-s + (0.960 − 1.66i)9-s + (−0.444 + 0.0471i)10-s + (1.16 − 0.675i)11-s + (0.358 + 1.67i)12-s − 1.03i·13-s + (0.790 − 0.612i)14-s + 0.764·15-s + (−0.811 − 0.584i)16-s + (−0.358 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.399 + 0.916i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09054 - 0.714359i\)
\(L(\frac12)\) \(\approx\) \(1.09054 - 0.714359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 + 0.832i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.64 + 0.0805i)T \)
good3 \( 1 + (2.56 - 1.48i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.72iT - 13T^{2} \)
17 \( 1 + (1.47 + 2.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.52 - 2.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.83 - 6.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.89iT - 29T^{2} \)
31 \( 1 + (1.50 + 2.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.99 - 1.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.55T + 41T^{2} \)
43 \( 1 - 7.13iT - 43T^{2} \)
47 \( 1 + (-2.36 + 4.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.38 - 1.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.55 - 3.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.30 - 1.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.18 - 5.30i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + (-4.82 - 8.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.77 + 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.41iT - 83T^{2} \)
89 \( 1 + (7.60 - 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−11.65165637933638459129087951955, −11.16065537154539553495911280625, −10.16714022402074348820926413306, −9.293841304637204597454816852881, −7.60969589272871086683271578709, −6.06931762446202041762468750912, −5.46045627541931411388586997661, −4.49520935727151422615028644459, −3.59408768062908437907763219525, −1.04793427698870851818778774202, 1.81999493773727935068074982536, 4.20988705245656277683920277997, 4.96532699939529828227615080016, 6.20579701461084691708814269543, 6.88179504834980508380553531512, 7.60444841416444126819326358316, 8.877321975345575533788752274176, 10.77092313971941316212740340212, 11.48986990120661049579340792137, 12.06922702325269204280357806101

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