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Properties

Label 2-280-280.19-c1-0-2
Degree $2$
Conductor $280$
Sign $-0.896 - 0.443i$
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  + (−1.38 − 0.274i)2-s + (1.60 + 2.77i)3-s + (1.84 + 0.761i)4-s + (−2.12 − 0.683i)5-s + (−1.46 − 4.29i)6-s + (−2.23 + 1.41i)7-s + (−2.35 − 1.56i)8-s + (−3.63 + 6.29i)9-s + (2.76 + 1.53i)10-s + (−1.23 − 2.14i)11-s + (0.849 + 6.35i)12-s + 2.90i·13-s + (3.49 − 1.34i)14-s + (−1.51 − 7.00i)15-s + (2.83 + 2.81i)16-s + (1.21 + 2.10i)17-s + ⋯
L(s)  = 1  + (−0.980 − 0.194i)2-s + (0.925 + 1.60i)3-s + (0.924 + 0.380i)4-s + (−0.952 − 0.305i)5-s + (−0.596 − 1.75i)6-s + (−0.845 + 0.534i)7-s + (−0.833 − 0.553i)8-s + (−1.21 + 2.09i)9-s + (0.874 + 0.484i)10-s + (−0.372 − 0.646i)11-s + (0.245 + 1.83i)12-s + 0.806i·13-s + (0.932 − 0.360i)14-s + (−0.390 − 1.80i)15-s + (0.709 + 0.704i)16-s + (0.294 + 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.896 - 0.443i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147721 + 0.631437i\)
\(L(\frac12)\) \(\approx\) \(0.147721 + 0.631437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.274i)T \)
5 \( 1 + (2.12 + 0.683i)T \)
7 \( 1 + (2.23 - 1.41i)T \)
good3 \( 1 + (-1.60 - 2.77i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.23 + 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.90iT - 13T^{2} \)
17 \( 1 + (-1.21 - 2.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.38 + 1.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 + 4.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.71iT - 29T^{2} \)
31 \( 1 + (-3.92 - 6.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.67 - 2.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.75iT - 41T^{2} \)
43 \( 1 - 0.763iT - 43T^{2} \)
47 \( 1 + (-2.86 - 1.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.474 - 0.821i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.41 - 3.70i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.145 + 0.251i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.12 + 2.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.87iT - 71T^{2} \)
73 \( 1 + (-0.915 - 1.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.69 - 0.978i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.61T + 83T^{2} \)
89 \( 1 + (-3.72 - 2.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−11.95932161519870110602247998592, −10.85456150225039152722073286008, −10.35244237964398887825176100585, −9.120197900749088371130500827681, −8.808159273703329014267091938526, −8.025622297605073338479227688876, −6.53667543230274124736835675922, −4.81683131391883276774439549070, −3.56824768394756844642936726653, −2.77724405106045400782052662480, 0.57721735410915795595710497761, 2.39397488560800820729681518224, 3.46856697285180007152716002761, 6.06077171500625683257811867909, 7.09370367478830573683046223888, 7.55777044935646790225827221077, 8.220361064868402815918727905082, 9.327506609236816966953719175802, 10.34773109839354087901718879147, 11.57749383369639891727923460588

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