Properties

Label 2-280-56.19-c1-0-5
Degree $2$
Conductor $280$
Sign $0.0638 - 0.997i$
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  + (−0.938 + 1.05i)2-s + (−0.219 + 0.126i)3-s + (−0.239 − 1.98i)4-s + (−0.5 + 0.866i)5-s + (0.0718 − 0.351i)6-s + (0.978 − 2.45i)7-s + (2.32 + 1.60i)8-s + (−1.46 + 2.54i)9-s + (−0.447 − 1.34i)10-s + (1.81 + 3.14i)11-s + (0.304 + 0.405i)12-s + 5.36·13-s + (1.68 + 3.34i)14-s − 0.253i·15-s + (−3.88 + 0.951i)16-s + (−4.46 + 2.58i)17-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (−0.126 + 0.0731i)3-s + (−0.119 − 0.992i)4-s + (−0.223 + 0.387i)5-s + (0.0293 − 0.143i)6-s + (0.369 − 0.929i)7-s + (0.822 + 0.569i)8-s + (−0.489 + 0.847i)9-s + (−0.141 − 0.424i)10-s + (0.547 + 0.947i)11-s + (0.0877 + 0.117i)12-s + 1.48·13-s + (0.449 + 0.893i)14-s − 0.0654i·15-s + (−0.971 + 0.237i)16-s + (−1.08 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0638 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0638 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.0638 - 0.997i$
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: \((2,\ 280,\ (\ :1/2),\ 0.0638 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636543 + 0.597113i\)
\(L(\frac12)\) \(\approx\) \(0.636543 + 0.597113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.938 - 1.05i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.978 + 2.45i)T \)
good3 \( 1 + (0.219 - 0.126i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.81 - 3.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.36T + 13T^{2} \)
17 \( 1 + (4.46 - 2.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.49 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.231 - 0.133i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.99iT - 29T^{2} \)
31 \( 1 + (-2.72 - 4.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.48 + 4.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 5.33T + 43T^{2} \)
47 \( 1 + (-2.26 + 3.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.09 + 1.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.83 + 3.94i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.63 - 4.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.963 - 1.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 + (-7.12 + 4.11i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.94 + 5.74i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.75iT - 83T^{2} \)
89 \( 1 + (-4.77 - 2.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.98iT - 97T^{2} \)
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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

−11.74254799266068381825224633014, −10.76616048072884045689355309527, −10.40082739695690897775898019380, −9.048814755912746603099392136791, −8.141216318251883676507963011743, −7.26743919727614206268664599464, −6.39050878818784027613569383935, −5.12908728533062653822869144734, −3.92005790049279766691342615719, −1.59613522216611778274127551136, 0.964523253517324158727950035630, 2.82787010853863755566966283362, 4.00498139098755459612521921211, 5.59620105184587432445121777658, 6.76061776064460250616295460273, 8.285424441145472428779990208183, 8.855374412889453490541688189343, 9.443371538638938916052035822841, 11.12188789323935986642281284499, 11.47448928858627397920297615487

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