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Properties

Label 2-280-56.19-c1-0-29
Degree $2$
Conductor $280$
Sign $-0.191 + 0.981i$
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  + (0.669 − 1.24i)2-s + (1.94 − 1.12i)3-s + (−1.10 − 1.66i)4-s + (−0.5 + 0.866i)5-s + (−0.0964 − 3.17i)6-s + (2.47 − 0.947i)7-s + (−2.81 + 0.257i)8-s + (1.02 − 1.78i)9-s + (0.743 + 1.20i)10-s + (0.656 + 1.13i)11-s + (−4.02 − 2.00i)12-s − 3.02·13-s + (0.473 − 3.71i)14-s + 2.24i·15-s + (−1.56 + 3.68i)16-s + (−0.313 + 0.181i)17-s + ⋯
L(s)  = 1  + (0.473 − 0.880i)2-s + (1.12 − 0.649i)3-s + (−0.551 − 0.834i)4-s + (−0.223 + 0.387i)5-s + (−0.0393 − 1.29i)6-s + (0.933 − 0.358i)7-s + (−0.995 + 0.0908i)8-s + (0.342 − 0.593i)9-s + (0.235 + 0.380i)10-s + (0.198 + 0.342i)11-s + (−1.16 − 0.579i)12-s − 0.838·13-s + (0.126 − 0.991i)14-s + 0.580i·15-s + (−0.391 + 0.920i)16-s + (−0.0760 + 0.0439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.191 + 0.981i$
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: \((2,\ 280,\ (\ :1/2),\ -0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33702 - 1.62345i\)
\(L(\frac12)\) \(\approx\) \(1.33702 - 1.62345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.669 + 1.24i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.47 + 0.947i)T \)
good3 \( 1 + (-1.94 + 1.12i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.656 - 1.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
17 \( 1 + (0.313 - 0.181i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.16 - 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.74 + 3.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.35iT - 29T^{2} \)
31 \( 1 + (-3.37 - 5.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 - 2.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.07iT - 41T^{2} \)
43 \( 1 + 5.01T + 43T^{2} \)
47 \( 1 + (-6.23 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.39 - 1.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.5 - 6.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.04 + 8.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.897 + 1.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (-7.83 + 4.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.89 + 4.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 + (1.99 + 1.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.74iT - 97T^{2} \)
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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

−11.85176533174357036326136818433, −10.68857291598885285988358274423, −9.866214839796621001386660790149, −8.680306823806479268620118127542, −7.84957345988921292968611521081, −6.83327232335454368099034715462, −5.17070942302301418808997587206, −3.98532069270125675567007411235, −2.74114935916115363312703918790, −1.66498923035021734309172565933, 2.66849595766888831062520319716, 4.00386942445977429398438048123, 4.81921313242587118814882747453, 5.99622675462267818793756241440, 7.69489340226522404672886839308, 8.063257971561765723764662168832, 9.131448041764528916233564847409, 9.709537233807456712662150082934, 11.46654039600738424453955420646, 12.17237733690818695938769530748

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