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Properties

Label 2-280-35.4-c1-0-4
Degree $2$
Conductor $280$
Sign $0.722 + 0.691i$
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  + (−2.20 − 1.27i)3-s + (−0.474 + 2.18i)5-s + (2.64 − 0.0988i)7-s + (1.73 + 3.00i)9-s + (1.95 − 3.38i)11-s − 3.47i·13-s + (3.82 − 4.21i)15-s + (4.40 + 2.54i)17-s + (−2.62 − 4.54i)19-s + (−5.95 − 3.14i)21-s + (5.21 − 3.00i)23-s + (−4.54 − 2.07i)25-s − 1.20i·27-s + 10.2·29-s + (−1.02 + 1.76i)31-s + ⋯
L(s)  = 1  + (−1.27 − 0.734i)3-s + (−0.212 + 0.977i)5-s + (0.999 − 0.0373i)7-s + (0.578 + 1.00i)9-s + (0.589 − 1.02i)11-s − 0.963i·13-s + (0.987 − 1.08i)15-s + (1.06 + 0.616i)17-s + (−0.602 − 1.04i)19-s + (−1.29 − 0.686i)21-s + (1.08 − 0.627i)23-s + (−0.909 − 0.414i)25-s − 0.231i·27-s + 1.90·29-s + (−0.183 + 0.317i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860525 - 0.345433i\)
\(L(\frac12)\) \(\approx\) \(0.860525 - 0.345433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.474 - 2.18i)T \)
7 \( 1 + (-2.64 + 0.0988i)T \)
good3 \( 1 + (2.20 + 1.27i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.95 + 3.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.47iT - 13T^{2} \)
17 \( 1 + (-4.40 - 2.54i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.62 + 4.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.21 + 3.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + (1.02 - 1.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.350 + 0.202i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 - 6.15iT - 43T^{2} \)
47 \( 1 + (0.416 - 0.240i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.403 - 0.233i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.15 - 8.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.04 + 6.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.59 + 2.07i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.28T + 71T^{2} \)
73 \( 1 + (-4.16 - 2.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.33 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.42iT - 83T^{2} \)
89 \( 1 + (-1.19 - 2.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.32iT - 97T^{2} \)
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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

−11.66391088740462988548383007872, −10.86627900856115852930834209118, −10.49059570782367550077227291501, −8.615986012825205093582891158569, −7.67358261994774557436988914930, −6.64744395742965512606844005843, −5.93501188522688208161298938267, −4.79134378074435806597073139304, −3.04266436890691023897152718302, −1.01774615897885914215854731857, 1.41303096049942312418014379683, 4.16322935840860066532020658967, 4.79938374297153179357857502388, 5.61227320937928869069602746511, 6.95901804290184985392350133053, 8.200270344601732547426345085596, 9.320017377783686256666764518091, 10.13516876447080174425628662097, 11.19679619602897441326807112234, 12.08089344384844454011935814489

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