↑

Properties

Label 2-280-280.59-c1-0-14
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

Related objects

Downloads

Learn more

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: $\chi_{280} (59, \cdot )$
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 \( 1 - 13.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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