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Properties

Label 2-280-35.4-c1-0-10
Degree $2$
Conductor $280$
Sign $0.340 + 0.940i$
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  + (0.650 + 0.375i)3-s + (−1.48 − 1.67i)5-s + (−0.543 − 2.58i)7-s + (−1.21 − 2.10i)9-s + (2.63 − 4.57i)11-s + 2.43i·13-s + (−0.340 − 1.64i)15-s + (5.30 + 3.06i)17-s + (−0.220 − 0.382i)19-s + (0.619 − 1.88i)21-s + (−2.75 + 1.59i)23-s + (−0.578 + 4.96i)25-s − 4.08i·27-s + 1.25·29-s + (0.322 − 0.558i)31-s + ⋯
L(s)  = 1  + (0.375 + 0.216i)3-s + (−0.664 − 0.746i)5-s + (−0.205 − 0.978i)7-s + (−0.405 − 0.703i)9-s + (0.795 − 1.37i)11-s + 0.675i·13-s + (−0.0878 − 0.424i)15-s + (1.28 + 0.742i)17-s + (−0.0506 − 0.0876i)19-s + (0.135 − 0.412i)21-s + (−0.575 + 0.332i)23-s + (−0.115 + 0.993i)25-s − 0.786i·27-s + 0.232·29-s + (0.0579 − 0.100i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.340 + 0.940i$
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: \((2,\ 280,\ (\ :1/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993564 - 0.697086i\)
\(L(\frac12)\) \(\approx\) \(0.993564 - 0.697086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 + 1.67i)T \)
7 \( 1 + (0.543 + 2.58i)T \)
good3 \( 1 + (-0.650 - 0.375i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.63 + 4.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.43iT - 13T^{2} \)
17 \( 1 + (-5.30 - 3.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.220 + 0.382i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.75 - 1.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (-0.322 + 0.558i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.31 - 5.37i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.90T + 41T^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (-6.52 + 3.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.89 + 1.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.07 - 7.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.73 - 3.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.14 - 2.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + (-4.61 - 2.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.70 - 4.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + (-4.30 - 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.6iT - 97T^{2} \)
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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

−11.79430717765565042781725580780, −10.78749055447444957066246014611, −9.628661173757034557707270706018, −8.749976507336793971658718692825, −8.035426771769327076952929003774, −6.77264191348723412858543591045, −5.62775344509476249621404971479, −3.95680865862494362173173432708, −3.55671819597998167158505519842, −0.954079798443945080853871904437, 2.26802971934802808053237024989, 3.35664280107565179892031776293, 4.88964580685946796732018104671, 6.15320622375633209717009757967, 7.38331062856054580345564080824, 7.993327208573173377258186753452, 9.194741807307735298430908137649, 10.12240814442401898176541360827, 11.19344059496537430605441165454, 12.19523339000153172844852565449

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