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Properties

Label 2-280-35.9-c1-0-2
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: $\chi_{280} (9, \cdot )$
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

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

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