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Properties

Label 2-280-56.19-c1-0-19
Degree $2$
Conductor $280$
Sign $0.145 + 0.989i$
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  + (−1.33 − 0.461i)2-s + (1.90 − 1.10i)3-s + (1.57 + 1.23i)4-s + (−0.5 + 0.866i)5-s + (−3.05 + 0.591i)6-s + (−0.584 − 2.58i)7-s + (−1.53 − 2.37i)8-s + (0.922 − 1.59i)9-s + (1.06 − 0.926i)10-s + (−2.90 − 5.03i)11-s + (4.35 + 0.620i)12-s + 4.83·13-s + (−0.410 + 3.71i)14-s + 2.20i·15-s + (0.953 + 3.88i)16-s + (3.78 − 2.18i)17-s + ⋯
L(s)  = 1  + (−0.945 − 0.326i)2-s + (1.10 − 0.635i)3-s + (0.786 + 0.617i)4-s + (−0.223 + 0.387i)5-s + (−1.24 + 0.241i)6-s + (−0.220 − 0.975i)7-s + (−0.542 − 0.840i)8-s + (0.307 − 0.532i)9-s + (0.337 − 0.293i)10-s + (−0.875 − 1.51i)11-s + (1.25 + 0.179i)12-s + 1.34·13-s + (−0.109 + 0.993i)14-s + 0.568i·15-s + (0.238 + 0.971i)16-s + (0.917 − 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.145 + 0.989i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.145 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849356 - 0.733910i\)
\(L(\frac12)\) \(\approx\) \(0.849356 - 0.733910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.461i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.584 + 2.58i)T \)
good3 \( 1 + (-1.90 + 1.10i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.90 + 5.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.83T + 13T^{2} \)
17 \( 1 + (-3.78 + 2.18i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 0.945i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.157 - 0.0911i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.38iT - 29T^{2} \)
31 \( 1 + (2.03 + 3.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.69 - 2.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 + 2.10T + 43T^{2} \)
47 \( 1 + (-0.946 + 1.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.54 - 4.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.35 - 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.89 - 8.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.04 - 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + (-7.34 + 4.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.5 - 6.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 + (5.11 + 2.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−11.23508862160880218748199060996, −10.78737869468617866257364451060, −9.660495952660362403357032885568, −8.538465158955042908474696129860, −7.947321226983739135699150527127, −7.24502014130417746160243611768, −6.02587287497276949214846065594, −3.51363097036294368873781338733, −2.97747495524414601673829046135, −1.12395427201998456090608245701, 2.04293097289518289546223315070, 3.36728972762881173311877552414, 5.01344747776955438268955559239, 6.22760584491044905467129913385, 7.73895123484484312262445810419, 8.307547349261119436081917757206, 9.255193213975097382900922657036, 9.752104506809048724195679478967, 10.77189369480286249814679496629, 12.01329007859125276862696943611

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