Properties

Label 2-280-35.12-c1-0-6
Degree $2$
Conductor $280$
Sign $0.950 + 0.311i$
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  + (0.447 − 1.66i)3-s + (0.849 + 2.06i)5-s + (2.48 + 0.900i)7-s + (0.0138 + 0.00800i)9-s + (−1.10 − 1.91i)11-s + (1.91 + 1.91i)13-s + (3.83 − 0.491i)15-s + (−4.19 − 1.12i)17-s + (1.80 − 3.12i)19-s + (2.61 − 3.74i)21-s + (−0.100 − 0.375i)23-s + (−3.55 + 3.51i)25-s + (3.68 − 3.68i)27-s − 6.62i·29-s + (0.897 − 0.518i)31-s + ⋯
L(s)  = 1  + (0.258 − 0.963i)3-s + (0.379 + 0.925i)5-s + (0.940 + 0.340i)7-s + (0.00462 + 0.00266i)9-s + (−0.333 − 0.578i)11-s + (0.530 + 0.530i)13-s + (0.989 − 0.127i)15-s + (−1.01 − 0.272i)17-s + (0.413 − 0.716i)19-s + (0.570 − 0.817i)21-s + (−0.0209 − 0.0783i)23-s + (−0.711 + 0.702i)25-s + (0.708 − 0.708i)27-s − 1.22i·29-s + (0.161 − 0.0930i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.950 + 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56921 - 0.250378i\)
\(L(\frac12)\) \(\approx\) \(1.56921 - 0.250378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.849 - 2.06i)T \)
7 \( 1 + (-2.48 - 0.900i)T \)
good3 \( 1 + (-0.447 + 1.66i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.10 + 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.91 - 1.91i)T + 13iT^{2} \)
17 \( 1 + (4.19 + 1.12i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.80 + 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.100 + 0.375i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.62iT - 29T^{2} \)
31 \( 1 + (-0.897 + 0.518i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.88 - 1.84i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.03iT - 41T^{2} \)
43 \( 1 + (8.37 - 8.37i)T - 43iT^{2} \)
47 \( 1 + (-1.48 - 5.52i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.07 + 1.62i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.71 + 6.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.84 - 4.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.70 + 10.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 + (3.69 - 13.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.38 + 3.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.77 + 7.77i)T + 83iT^{2} \)
89 \( 1 + (5.20 - 9.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.1 + 12.1i)T - 97iT^{2} \)
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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

−11.52159385851666583234815641218, −11.21989210257299369888557883212, −9.974480954703589416327431614155, −8.723592093227135331208378904291, −7.87149358446638535164375687367, −6.91175047228433696759453003779, −6.08947073761373732201973651674, −4.65307091939501937103854502202, −2.84046960314731310502326437431, −1.74795779201730644723654971815, 1.67607829594830039408371999528, 3.71877187158065172019015191397, 4.71504663152639090170709754054, 5.46653119171160423220267684598, 7.11671281864190878216103914581, 8.405064163754872254318043552554, 8.964936906730054977628134580023, 10.15669131193632359326935789819, 10.62812541499432311080451157344, 11.91574950399377350658214878024

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