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Properties

Label 2-280-280.67-c1-0-28
Degree $2$
Conductor $280$
Sign $-0.392 + 0.919i$
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  + (−0.784 − 1.17i)2-s + (−0.0488 − 0.182i)3-s + (−0.768 + 1.84i)4-s + (−0.546 − 2.16i)5-s + (−0.176 + 0.200i)6-s + (1.82 + 1.91i)7-s + (2.77 − 0.543i)8-s + (2.56 − 1.48i)9-s + (−2.12 + 2.34i)10-s + (1.05 − 1.83i)11-s + (0.374 + 0.0499i)12-s + (−3.49 − 3.49i)13-s + (0.825 − 3.64i)14-s + (−0.368 + 0.205i)15-s + (−2.81 − 2.83i)16-s + (−3.56 + 0.955i)17-s + ⋯
L(s)  = 1  + (−0.554 − 0.831i)2-s + (−0.0282 − 0.105i)3-s + (−0.384 + 0.923i)4-s + (−0.244 − 0.969i)5-s + (−0.0719 + 0.0818i)6-s + (0.689 + 0.724i)7-s + (0.981 − 0.192i)8-s + (0.855 − 0.494i)9-s + (−0.671 + 0.741i)10-s + (0.318 − 0.551i)11-s + (0.108 + 0.0144i)12-s + (−0.969 − 0.969i)13-s + (0.220 − 0.975i)14-s + (−0.0951 + 0.0530i)15-s + (−0.704 − 0.709i)16-s + (−0.864 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.392 + 0.919i$
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: \((2,\ 280,\ (\ :1/2),\ -0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515452 - 0.780577i\)
\(L(\frac12)\) \(\approx\) \(0.515452 - 0.780577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.784 + 1.17i)T \)
5 \( 1 + (0.546 + 2.16i)T \)
7 \( 1 + (-1.82 - 1.91i)T \)
good3 \( 1 + (0.0488 + 0.182i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.05 + 1.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.49 + 3.49i)T + 13iT^{2} \)
17 \( 1 + (3.56 - 0.955i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.681 + 0.393i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.28 + 8.51i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + (-3.12 - 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.30 + 0.885i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.47T + 41T^{2} \)
43 \( 1 + (1.31 - 1.31i)T - 43iT^{2} \)
47 \( 1 + (-4.34 - 1.16i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.40 - 0.376i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.46 - 3.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.644 + 0.372i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.8 + 3.18i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.25iT - 71T^{2} \)
73 \( 1 + (-2.92 - 10.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.89 - 6.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.14 - 8.14i)T - 83iT^{2} \)
89 \( 1 + (-4.78 + 2.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.2 + 12.2i)T + 97iT^{2} \)
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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

−11.68758895641945783935695432780, −10.62588771616542697087721802884, −9.651948552448950119317513697517, −8.630887240103911626533877820375, −8.206616016959901441157077070495, −6.83959033817364079991566598284, −5.09831980995444256049886865799, −4.22913762961396104705094719972, −2.55426001208039613614387194097, −0.941676053560091168031342923179, 1.86376434766365032152836098962, 4.17175652338659471683382367204, 5.00280501284917289968467787704, 6.78827510562164317372478412422, 7.14407475999083261019026160673, 7.997135395932272049913157451962, 9.437076430577400392773692589619, 10.12404993009510337687207331956, 10.98195687437057827473289136048, 11.86079123870686581728480422848

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