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Properties

Label 2-280-280.67-c1-0-27
Degree $2$
Conductor $280$
Sign $0.999 + 0.0305i$
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  + (−0.942 + 1.05i)2-s + (0.438 + 1.63i)3-s + (−0.223 − 1.98i)4-s + (−0.615 − 2.14i)5-s + (−2.13 − 1.07i)6-s + (−0.885 − 2.49i)7-s + (2.30 + 1.63i)8-s + (0.114 − 0.0663i)9-s + (2.84 + 1.37i)10-s + (2.94 − 5.09i)11-s + (3.15 − 1.23i)12-s + (1.49 + 1.49i)13-s + (3.46 + 1.41i)14-s + (3.24 − 1.94i)15-s + (−3.90 + 0.888i)16-s + (−5.43 + 1.45i)17-s + ⋯
L(s)  = 1  + (−0.666 + 0.745i)2-s + (0.253 + 0.944i)3-s + (−0.111 − 0.993i)4-s + (−0.275 − 0.961i)5-s + (−0.872 − 0.440i)6-s + (−0.334 − 0.942i)7-s + (0.815 + 0.578i)8-s + (0.0383 − 0.0221i)9-s + (0.900 + 0.435i)10-s + (0.887 − 1.53i)11-s + (0.910 − 0.356i)12-s + (0.414 + 0.414i)13-s + (0.925 + 0.378i)14-s + (0.838 − 0.503i)15-s + (−0.975 + 0.222i)16-s + (−1.31 + 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0305i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.999 + 0.0305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922457 - 0.0140981i\)
\(L(\frac12)\) \(\approx\) \(0.922457 - 0.0140981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.942 - 1.05i)T \)
5 \( 1 + (0.615 + 2.14i)T \)
7 \( 1 + (0.885 + 2.49i)T \)
good3 \( 1 + (-0.438 - 1.63i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.94 + 5.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.49 - 1.49i)T + 13iT^{2} \)
17 \( 1 + (5.43 - 1.45i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.10 + 1.79i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.669 + 2.49i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.02T + 29T^{2} \)
31 \( 1 + (-1.05 - 0.609i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-10.0 - 2.68i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 + (-7.70 + 7.70i)T - 43iT^{2} \)
47 \( 1 + (4.15 + 1.11i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.68 - 1.25i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.51 - 1.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.34 + 3.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.27 - 2.21i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.68iT - 71T^{2} \)
73 \( 1 + (-0.400 - 1.49i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.03 - 1.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.700 - 0.700i)T - 83iT^{2} \)
89 \( 1 + (3.21 - 1.85i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.71 - 2.71i)T + 97iT^{2} \)
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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

−11.37376012297603350303566160340, −10.81197654960611264753711993537, −9.542801067271453026598494773775, −9.070847348920012419072653600966, −8.293932646386039121642231217204, −6.99038647642239884150777186063, −5.96208998832051717623157820618, −4.54812342277155048573587510654, −3.81838238155826385098345550388, −0.927688389244265385437738562220, 1.82486321722199469349788890691, 2.78637844522795517988085096119, 4.23557588589277677465555928959, 6.33531221211754084826436066026, 7.23642428325418609945532806405, 7.87558298089095110643837554062, 9.218356673571249033783879836208, 9.814827479193164464562737563739, 11.12852673835052127208201912519, 11.78019453347314684105559642259

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