Properties

Label 2-280-35.17-c1-0-7
Degree $2$
Conductor $280$
Sign $-0.215 + 0.976i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 + 0.364i)3-s + (−2.13 − 0.666i)5-s + (2.59 − 0.501i)7-s + (−0.882 + 0.509i)9-s + (1.86 − 3.23i)11-s + (−4.55 − 4.55i)13-s + (3.14 + 0.129i)15-s + (−1.58 − 5.91i)17-s + (−0.616 − 1.06i)19-s + (−3.34 + 1.62i)21-s + (−2.69 − 0.721i)23-s + (4.11 + 2.84i)25-s + (3.99 − 3.99i)27-s − 2.82i·29-s + (5.94 + 3.43i)31-s + ⋯
L(s)  = 1  + (−0.784 + 0.210i)3-s + (−0.954 − 0.298i)5-s + (0.981 − 0.189i)7-s + (−0.294 + 0.169i)9-s + (0.563 − 0.975i)11-s + (−1.26 − 1.26i)13-s + (0.811 + 0.0333i)15-s + (−0.384 − 1.43i)17-s + (−0.141 − 0.244i)19-s + (−0.730 + 0.355i)21-s + (−0.561 − 0.150i)23-s + (0.822 + 0.569i)25-s + (0.769 − 0.769i)27-s − 0.525i·29-s + (1.06 + 0.616i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383685 - 0.477629i\)
\(L(\frac12)\) \(\approx\) \(0.383685 - 0.477629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 0.666i)T \)
7 \( 1 + (-2.59 + 0.501i)T \)
good3 \( 1 + (1.35 - 0.364i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.86 + 3.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.55 + 4.55i)T + 13iT^{2} \)
17 \( 1 + (1.58 + 5.91i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.616 + 1.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.69 + 0.721i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (-5.94 - 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.01 - 7.52i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.56iT - 41T^{2} \)
43 \( 1 + (1.95 - 1.95i)T - 43iT^{2} \)
47 \( 1 + (11.1 + 2.98i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.53 + 9.44i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.916 - 1.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.43 - 1.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.93 + 2.12i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.570T + 71T^{2} \)
73 \( 1 + (2.03 - 0.546i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.47 + 5.47i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.80 - 8.80i)T + 83iT^{2} \)
89 \( 1 + (3.00 + 5.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.70 + 3.70i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.65065884236879887896053125695, −10.94712074710233610258016529880, −9.882877179274273785321257504485, −8.416516767332709150386270226914, −7.908730446444391653473948593096, −6.61952406672088782470777026032, −5.14697953168209659939518457937, −4.71066923287425649686244432964, −3.01318990091861485766580687487, −0.50888744727726992361461133672, 1.96854755820331507335958124198, 4.04332507734869933223641459583, 4.86947171551509003708787449709, 6.30607201154255947955869654862, 7.16856706759077226186639027343, 8.151646383231464712789098324647, 9.236131351706964519166000331298, 10.54616820373705040462014751447, 11.38895717853583820446534036821, 12.08605042738754502582224071925

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