Properties

Label 2-280-56.3-c1-0-21
Degree $2$
Conductor $280$
Sign $-0.911 + 0.411i$
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 − 1.34i)2-s + (−0.219 − 0.126i)3-s + (−1.59 + 1.20i)4-s + (0.5 + 0.866i)5-s + (−0.0718 + 0.351i)6-s + (−0.978 − 2.45i)7-s + (2.32 + 1.60i)8-s + (−1.46 − 2.54i)9-s + (0.938 − 1.05i)10-s + (1.81 − 3.14i)11-s + (0.503 − 0.0606i)12-s − 5.36·13-s + (−2.85 + 2.41i)14-s − 0.253i·15-s + (1.11 − 3.84i)16-s + (−4.46 − 2.58i)17-s + ⋯
L(s)  = 1  + (−0.316 − 0.948i)2-s + (−0.126 − 0.0731i)3-s + (−0.799 + 0.600i)4-s + (0.223 + 0.387i)5-s + (−0.0293 + 0.143i)6-s + (−0.369 − 0.929i)7-s + (0.822 + 0.569i)8-s + (−0.489 − 0.847i)9-s + (0.296 − 0.334i)10-s + (0.547 − 0.947i)11-s + (0.145 − 0.0175i)12-s − 1.48·13-s + (−0.764 + 0.644i)14-s − 0.0654i·15-s + (0.279 − 0.960i)16-s + (−1.08 − 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.911 + 0.411i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150825 - 0.700792i\)
\(L(\frac12)\) \(\approx\) \(0.150825 - 0.700792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.447 + 1.34i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.978 + 2.45i)T \)
good3 \( 1 + (0.219 + 0.126i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.81 + 3.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.36T + 13T^{2} \)
17 \( 1 + (4.46 + 2.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.49 + 3.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.231 - 0.133i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.99iT - 29T^{2} \)
31 \( 1 + (2.72 - 4.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.48 + 4.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 5.33T + 43T^{2} \)
47 \( 1 + (2.26 + 3.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.09 + 1.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.83 - 3.94i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.63 - 4.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.963 + 1.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 + (-7.12 - 4.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.94 + 5.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.75iT - 83T^{2} \)
89 \( 1 + (-4.77 + 2.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.98iT - 97T^{2} \)
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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

−11.38726363612045549214489433404, −10.64985337610422408215615280941, −9.489632804708997640536935211910, −9.088380852849057594062066371106, −7.52617004318667464836390276511, −6.69091484755473421776255496414, −5.11589381011132562653189357052, −3.72032610747998873028692527710, −2.72438895984119551717562103637, −0.60692498858292576358813387633, 2.19646090087921981503384989402, 4.46195461879194235281937733291, 5.34484115409116251918137768498, 6.27295109177770215809537198266, 7.47830203769744411303581663441, 8.338074697534452851934021887176, 9.567978487237834336658942067236, 9.777571571831982639178232732952, 11.33943735055674215710871322486, 12.41304662274761788374140942846

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