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Properties

Label 2-280-280.19-c1-0-19
Degree $2$
Conductor $280$
Sign $0.999 - 0.0429i$
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  + (−1.30 + 0.549i)2-s + (0.833 + 1.44i)3-s + (1.39 − 1.43i)4-s + (−0.660 − 2.13i)5-s + (−1.88 − 1.42i)6-s + (2.56 − 0.635i)7-s + (−1.02 + 2.63i)8-s + (0.110 − 0.191i)9-s + (2.03 + 2.42i)10-s + (−0.395 − 0.684i)11-s + (3.23 + 0.819i)12-s − 6.84i·13-s + (−2.99 + 2.24i)14-s + (2.53 − 2.73i)15-s + (−0.107 − 3.99i)16-s + (1.46 + 2.53i)17-s + ⋯
L(s)  = 1  + (−0.921 + 0.388i)2-s + (0.481 + 0.833i)3-s + (0.697 − 0.716i)4-s + (−0.295 − 0.955i)5-s + (−0.767 − 0.580i)6-s + (0.970 − 0.240i)7-s + (−0.364 + 0.931i)8-s + (0.0367 − 0.0637i)9-s + (0.643 + 0.765i)10-s + (−0.119 − 0.206i)11-s + (0.932 + 0.236i)12-s − 1.89i·13-s + (−0.800 + 0.598i)14-s + (0.654 − 0.706i)15-s + (−0.0267 − 0.999i)16-s + (0.354 + 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0429i$
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: \((2,\ 280,\ (\ :1/2),\ 0.999 - 0.0429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04783 + 0.0225294i\)
\(L(\frac12)\) \(\approx\) \(1.04783 + 0.0225294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.549i)T \)
5 \( 1 + (0.660 + 2.13i)T \)
7 \( 1 + (-2.56 + 0.635i)T \)
good3 \( 1 + (-0.833 - 1.44i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.395 + 0.684i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.84iT - 13T^{2} \)
17 \( 1 + (-1.46 - 2.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.96 - 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.946 - 1.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.46iT - 29T^{2} \)
31 \( 1 + (-2.87 - 4.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.59 - 4.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.71iT - 41T^{2} \)
43 \( 1 + 9.52iT - 43T^{2} \)
47 \( 1 + (0.185 + 0.106i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.21 + 5.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.430 + 0.248i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.87 - 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.83 - 1.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.40iT - 71T^{2} \)
73 \( 1 + (-3.18 - 5.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.25 + 3.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + (9.86 + 5.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−11.65987759878470598639341600773, −10.45811684163759902282069682576, −10.03088245539759912394327017952, −8.742422678735811106865080819680, −8.302996438919922972397678569198, −7.45377394661045766004107902575, −5.68414028958740430909865547384, −4.85502956177061517456022697746, −3.34443133627746884524850641137, −1.18471223395149771939403103968, 1.76450000467430828496300262785, 2.66812152855321996101003691689, 4.34359730327973815130053749350, 6.43474583046054442419558150258, 7.34345419067144392441274086278, 7.84418540179690261184875776035, 8.916181488284315576194881867405, 9.906866371087316252563246616811, 11.11838467331399342417721977808, 11.60075293057772622067220236439

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