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Properties

Label 2-280-56.19-c1-0-13
Degree $2$
Conductor $280$
Sign $0.393 - 0.919i$
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  + (0.617 + 1.27i)2-s + (0.725 − 0.418i)3-s + (−1.23 + 1.57i)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 + (1.41 + 0.101i)10-s + (2.98 + 5.17i)11-s + (−0.239 + 1.65i)12-s + 2.87·13-s + (2.96 + 2.28i)14-s − 0.837i·15-s + (−0.938 − 3.88i)16-s + (2.07 − 1.19i)17-s + ⋯
L(s)  = 1  + (0.436 + 0.899i)2-s + (0.418 − 0.241i)3-s + (−0.618 + 0.785i)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 + (0.446 + 0.0320i)10-s + (0.901 + 1.56i)11-s + (−0.0691 + 0.478i)12-s + 0.797·13-s + (0.791 + 0.611i)14-s − 0.216i·15-s + (−0.234 − 0.972i)16-s + (0.503 − 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.393 - 0.919i$
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: \((2,\ 280,\ (\ :1/2),\ 0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53836 + 1.01533i\)
\(L(\frac12)\) \(\approx\) \(1.53836 + 1.01533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.617 - 1.27i)T \)
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

−12.27810663465466470901720581314, −11.30359916329358342740155313066, −9.914843234269060896208527718030, −8.824090804536834643076766488937, −8.043323268972692155512998284051, −7.25039391586730693093487862753, −6.09719161369045776974273999724, −4.81525724209475129189929989273, −4.06556799336125607549339785076, −2.04055627697121382953246188840, 1.57970352403114212948391331782, 3.22870750116510146703937233189, 3.95154216209316274674294667003, 5.64989501576623078522202798299, 6.24802005341205536114977448415, 8.401140287414537627012556441379, 8.775362414810370662802576861880, 9.928802983833735658751815100546, 11.01034988198793393842957402913, 11.53957362074380319314852722333

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