Properties

Label 2-280-56.3-c1-0-3
Degree $2$
Conductor $280$
Sign $0.510 - 0.859i$
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  + (0.314 + 1.37i)2-s + (−2.66 − 1.54i)3-s + (−1.80 + 0.867i)4-s + (−0.5 − 0.866i)5-s + (1.28 − 4.16i)6-s + (2.53 + 0.754i)7-s + (−1.76 − 2.21i)8-s + (3.24 + 5.61i)9-s + (1.03 − 0.961i)10-s + (−1.64 + 2.84i)11-s + (6.14 + 0.459i)12-s + 6.72·13-s + (−0.241 + 3.73i)14-s + 3.08i·15-s + (2.49 − 3.12i)16-s + (3.32 + 1.91i)17-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−1.54 − 0.889i)3-s + (−0.900 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.524 − 1.69i)6-s + (0.958 + 0.285i)7-s + (−0.623 − 0.781i)8-s + (1.08 + 1.87i)9-s + (0.327 − 0.304i)10-s + (−0.495 + 0.857i)11-s + (1.77 + 0.132i)12-s + 1.86·13-s + (−0.0645 + 0.997i)14-s + 0.795i·15-s + (0.623 − 0.781i)16-s + (0.806 + 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.510 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739812 + 0.420942i\)
\(L(\frac12)\) \(\approx\) \(0.739812 + 0.420942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.314 - 1.37i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.53 - 0.754i)T \)
good3 \( 1 + (2.66 + 1.54i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.64 - 2.84i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.72T + 13T^{2} \)
17 \( 1 + (-3.32 - 1.91i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.618 - 0.357i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 0.631i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.04iT - 29T^{2} \)
31 \( 1 + (-0.0335 + 0.0581i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.498 - 0.287i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.230iT - 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (4.23 + 7.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.16 - 1.25i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.986 - 0.569i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0888 + 0.153i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.92 - 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (3.89 + 2.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.66 + 5.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.24iT - 83T^{2} \)
89 \( 1 + (13.3 - 7.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.76iT - 97T^{2} \)
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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

−12.20790467060897527467789337669, −11.30792225396996279882440620233, −10.37452246599878467577566616845, −8.689981749213259155728188883829, −7.87709325425899044762746533403, −7.00759385445740135983460282098, −5.89631164662796711275605412493, −5.32516598425902442881756481041, −4.24117192629289439927953729114, −1.28284863937573339702463316464, 0.949549166586966896043797722723, 3.44128681027363936189780655671, 4.40139497572390530195113529973, 5.45499069355111914560049664036, 6.17236286136351737640582703812, 8.018642790998201682262839269566, 9.225148031369968756176620504385, 10.43320021716444051652939088313, 10.96257154357603390372769614584, 11.32363648694394231131039651551

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