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Properties

Label 2-280-280.19-c1-0-32
Degree $2$
Conductor $280$
Sign $0.743 + 0.668i$
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  + (−0.0226 + 1.41i)2-s + (−0.0769 − 0.133i)3-s + (−1.99 − 0.0639i)4-s + (0.159 − 2.23i)5-s + (0.190 − 0.105i)6-s + (−2.33 − 1.23i)7-s + (0.135 − 2.82i)8-s + (1.48 − 2.57i)9-s + (3.15 + 0.275i)10-s + (−0.717 − 1.24i)11-s + (0.145 + 0.271i)12-s − 1.09i·13-s + (1.80 − 3.27i)14-s + (−0.309 + 0.150i)15-s + (3.99 + 0.255i)16-s + (1.65 + 2.86i)17-s + ⋯
L(s)  = 1  + (−0.0159 + 0.999i)2-s + (−0.0444 − 0.0769i)3-s + (−0.999 − 0.0319i)4-s + (0.0712 − 0.997i)5-s + (0.0776 − 0.0432i)6-s + (−0.883 − 0.467i)7-s + (0.0479 − 0.998i)8-s + (0.496 − 0.859i)9-s + (0.996 + 0.0871i)10-s + (−0.216 − 0.374i)11-s + (0.0419 + 0.0783i)12-s − 0.302i·13-s + (0.481 − 0.876i)14-s + (−0.0799 + 0.0388i)15-s + (0.997 + 0.0639i)16-s + (0.401 + 0.695i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.743 + 0.668i$
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: \((2,\ 280,\ (\ :1/2),\ 0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841270 - 0.322683i\)
\(L(\frac12)\) \(\approx\) \(0.841270 - 0.322683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0226 - 1.41i)T \)
5 \( 1 + (-0.159 + 2.23i)T \)
7 \( 1 + (2.33 + 1.23i)T \)
good3 \( 1 + (0.0769 + 0.133i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.717 + 1.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.09iT - 13T^{2} \)
17 \( 1 + (-1.65 - 2.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.25 + 1.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.01 + 1.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.06iT - 29T^{2} \)
31 \( 1 + (-4.92 - 8.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 7.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.08iT - 41T^{2} \)
43 \( 1 + 2.12iT - 43T^{2} \)
47 \( 1 + (6.16 + 3.55i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.25 - 3.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.68 - 3.86i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.87 + 6.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.0 + 6.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 + (4.82 + 8.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.80 - 1.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.80T + 83T^{2} \)
89 \( 1 + (-9.61 - 5.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−12.21165595140367800925539705376, −10.45961287584112202130297603719, −9.618515189476445445070930994508, −8.795843637694863997358221561242, −7.87904809466197772746886843624, −6.66784594920565991726364935426, −5.95347029005258900470868713046, −4.64596771061324087912278677224, −3.61638871697294533929998924042, −0.71319723486794995768649282751, 2.17135741907940651310737208038, 3.19672920753528797106956666249, 4.52885122437747625533372802887, 5.85041404565810038111776261482, 7.12316079207834037172563423447, 8.251959545183539105307342850934, 9.679836340315026000457738348271, 10.01122699922995566134536672623, 11.02301230087470256539751931724, 11.81854471517376375860192967653

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