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Properties

Label 2-280-280.107-c1-0-28
Degree $2$
Conductor $280$
Sign $0.581 - 0.813i$
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  + (1.27 + 0.605i)2-s + (2.01 + 0.540i)3-s + (1.26 + 1.54i)4-s + (−2.01 + 0.975i)5-s + (2.25 + 1.91i)6-s + (2.04 − 1.67i)7-s + (0.683 + 2.74i)8-s + (1.17 + 0.679i)9-s + (−3.16 + 0.0288i)10-s + (−2.35 − 4.08i)11-s + (1.71 + 3.80i)12-s + (−3.68 − 3.68i)13-s + (3.63 − 0.898i)14-s + (−4.58 + 0.879i)15-s + (−0.787 + 3.92i)16-s + (−1.60 + 5.98i)17-s + ⋯
L(s)  = 1  + (0.903 + 0.427i)2-s + (1.16 + 0.311i)3-s + (0.633 + 0.773i)4-s + (−0.899 + 0.436i)5-s + (0.918 + 0.780i)6-s + (0.774 − 0.632i)7-s + (0.241 + 0.970i)8-s + (0.392 + 0.226i)9-s + (−0.999 + 0.00913i)10-s + (−0.711 − 1.23i)11-s + (0.496 + 1.09i)12-s + (−1.02 − 1.02i)13-s + (0.970 − 0.240i)14-s + (−1.18 + 0.227i)15-s + (−0.196 + 0.980i)16-s + (−0.389 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29851 + 1.18186i\)
\(L(\frac12)\) \(\approx\) \(2.29851 + 1.18186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.605i)T \)
5 \( 1 + (2.01 - 0.975i)T \)
7 \( 1 + (-2.04 + 1.67i)T \)
good3 \( 1 + (-2.01 - 0.540i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.35 + 4.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.68 + 3.68i)T + 13iT^{2} \)
17 \( 1 + (1.60 - 5.98i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.62 - 1.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.38 + 0.371i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + (0.111 - 0.0644i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.568 + 2.12i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 + (-2.05 + 2.05i)T - 43iT^{2} \)
47 \( 1 + (-2.96 - 11.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.758 + 2.82i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (11.1 - 6.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.49 + 3.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.95 - 7.28i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.96iT - 71T^{2} \)
73 \( 1 + (0.768 + 0.205i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.00532 - 0.00922i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.27 + 4.27i)T - 83iT^{2} \)
89 \( 1 + (6.15 + 3.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.22 - 8.22i)T + 97iT^{2} \)
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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

−12.17236496459864710096162676833, −11.02452643831205407338021619354, −10.44503847379411088384723616897, −8.615710426221246300665177031345, −7.953032879178960941114012617883, −7.47746176711869965618299350443, −5.89404219583514403674990938825, −4.56127863443733577699376528019, −3.54309005307845225583995274412, −2.74359869405135043312327316375, 2.03113414260463747403842270554, 2.92565705432274175796351378512, 4.57854980063634741645079531185, 5.03283547977139566007368915376, 7.15168893731942897168937357744, 7.60090721895865548857384237297, 8.923140447976982545103290689480, 9.673603624637139141437863381250, 11.19350651666974535887936219889, 11.95881566483723702955439832158

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