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Properties

Label 2-280-280.19-c1-0-34
Degree $2$
Conductor $280$
Sign $-0.986 + 0.166i$
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  + (−1.20 − 0.743i)2-s + (−0.818 − 1.41i)3-s + (0.894 + 1.78i)4-s + (−0.951 − 2.02i)5-s + (−0.0691 + 2.31i)6-s + (2.64 + 0.148i)7-s + (0.253 − 2.81i)8-s + (0.158 − 0.274i)9-s + (−0.359 + 3.14i)10-s + (−2.19 − 3.79i)11-s + (1.80 − 2.73i)12-s + 2.75i·13-s + (−3.06 − 2.14i)14-s + (−2.09 + 3.00i)15-s + (−2.39 + 3.20i)16-s + (−0.648 − 1.12i)17-s + ⋯
L(s)  = 1  + (−0.850 − 0.525i)2-s + (−0.472 − 0.818i)3-s + (0.447 + 0.894i)4-s + (−0.425 − 0.904i)5-s + (−0.0282 + 0.945i)6-s + (0.998 + 0.0561i)7-s + (0.0895 − 0.995i)8-s + (0.0529 − 0.0916i)9-s + (−0.113 + 0.993i)10-s + (−0.661 − 1.14i)11-s + (0.520 − 0.789i)12-s + 0.764i·13-s + (−0.819 − 0.572i)14-s + (−0.539 + 0.776i)15-s + (−0.599 + 0.800i)16-s + (−0.157 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.986 + 0.166i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.986 + 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0492389 - 0.588679i\)
\(L(\frac12)\) \(\approx\) \(0.0492389 - 0.588679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 0.743i)T \)
5 \( 1 + (0.951 + 2.02i)T \)
7 \( 1 + (-2.64 - 0.148i)T \)
good3 \( 1 + (0.818 + 1.41i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.19 + 3.79i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75iT - 13T^{2} \)
17 \( 1 + (0.648 + 1.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.86 + 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.136 + 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.84iT - 29T^{2} \)
31 \( 1 + (2.27 + 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.94 + 6.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.95iT - 41T^{2} \)
43 \( 1 + 4.46iT - 43T^{2} \)
47 \( 1 + (7.22 + 4.17i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.40 - 9.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.31 + 1.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.73 + 8.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.32 + 4.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.42iT - 71T^{2} \)
73 \( 1 + (-1.84 - 3.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.3 - 7.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (3.15 + 1.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0199T + 97T^{2} \)
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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

−11.29196930887706106107101549203, −10.90556132740305133786137654476, −9.241610304705778282807994150585, −8.565606043591352057356146173386, −7.72770692022136944321908465155, −6.76718722212764459749247212314, −5.33114627470344333417210492115, −3.92177970680795957340979873496, −1.97791884735171481440948363572, −0.62114029389852271645308701924, 2.22425363104839276358287977494, 4.32757265195809479545606920887, 5.27442559787118564254222359547, 6.50342607738411699899258913527, 7.75188457995039310115544612018, 8.125835978148950468797429071084, 9.839072011485513413421373653617, 10.31514963357035518425632958870, 11.01634863378028546397357427983, 11.77746467595487856012090483203

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