Properties

Label 2-280-56.19-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.0260 - 0.999i$
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  + (−1.41 + 0.101i)2-s + (0.725 − 0.418i)3-s + (1.97 − 0.285i)4-s + (−0.5 + 0.866i)5-s + (−0.981 + 0.664i)6-s + (−2.36 + 1.17i)7-s + (−2.76 + 0.603i)8-s + (−1.14 + 1.99i)9-s + (0.617 − 1.27i)10-s + (2.98 + 5.17i)11-s + (1.31 − 1.03i)12-s − 2.87·13-s + (3.22 − 1.90i)14-s + 0.837i·15-s + (3.83 − 1.13i)16-s + (2.07 − 1.19i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0716i)2-s + (0.418 − 0.241i)3-s + (0.989 − 0.142i)4-s + (−0.223 + 0.387i)5-s + (−0.400 + 0.271i)6-s + (−0.895 + 0.445i)7-s + (−0.976 + 0.213i)8-s + (−0.382 + 0.663i)9-s + (0.195 − 0.402i)10-s + (0.901 + 1.56i)11-s + (0.380 − 0.299i)12-s − 0.797·13-s + (0.861 − 0.508i)14-s + 0.216i·15-s + (0.959 − 0.282i)16-s + (0.503 − 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0260 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0260 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0260 - 0.999i$
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: \((2,\ 280,\ (\ :1/2),\ -0.0260 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486119 + 0.498959i\)
\(L(\frac12)\) \(\approx\) \(0.486119 + 0.498959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.101i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.36 - 1.17i)T \)
good3 \( 1 + (-0.725 + 0.418i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.98 - 5.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.87T + 13T^{2} \)
17 \( 1 + (-2.07 + 1.19i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.05 + 2.34i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.81 - 3.35i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.31iT - 29T^{2} \)
31 \( 1 + (-4.34 - 7.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.26 + 1.88i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.79iT - 41T^{2} \)
43 \( 1 + 6.73T + 43T^{2} \)
47 \( 1 + (-0.874 + 1.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0994 - 0.0574i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.61 + 1.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.40 + 7.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.83 + 4.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (-8.69 + 5.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.9 - 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−12.09225265889645526947764406975, −10.99361393292940396546379275309, −9.957811398227022503352268926854, −9.294789404598324336989021785657, −8.355602224306489713264548550240, −7.08386529459651135719377546528, −6.81712521085290408567558656543, −5.12677674547066563213122331305, −3.15066000821530314290024920336, −2.05802643825766961393773798321, 0.67567116388344353341435094031, 2.93454730331695101157582328720, 3.87811434054119976058728907365, 5.99927075364677043043810680777, 6.72778066524225602001765338059, 8.115107390750230248742988674076, 8.814351447999536937972024111214, 9.570066324979161766113567883899, 10.44993535868880592778568438879, 11.54324969233495300007010851443

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