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Properties

Label 2-280-280.19-c1-0-3
Degree $2$
Conductor $280$
Sign $0.841 - 0.540i$
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  + (−1.36 + 0.359i)2-s + (−1.46 − 2.53i)3-s + (1.74 − 0.984i)4-s + (−1.36 + 1.77i)5-s + (2.91 + 2.94i)6-s + (1.45 + 2.21i)7-s + (−2.02 + 1.97i)8-s + (−2.79 + 4.83i)9-s + (1.22 − 2.91i)10-s + (−1.55 − 2.68i)11-s + (−5.04 − 2.97i)12-s + 1.62i·13-s + (−2.78 − 2.49i)14-s + (6.49 + 0.865i)15-s + (2.06 − 3.42i)16-s + (2.52 + 4.37i)17-s + ⋯
L(s)  = 1  + (−0.967 + 0.254i)2-s + (−0.845 − 1.46i)3-s + (0.870 − 0.492i)4-s + (−0.610 + 0.792i)5-s + (1.19 + 1.20i)6-s + (0.549 + 0.835i)7-s + (−0.716 + 0.697i)8-s + (−0.931 + 1.61i)9-s + (0.388 − 0.921i)10-s + (−0.467 − 0.810i)11-s + (−1.45 − 0.859i)12-s + 0.451i·13-s + (−0.744 − 0.668i)14-s + (1.67 + 0.223i)15-s + (0.515 − 0.856i)16-s + (0.612 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.841 - 0.540i$
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: \((2,\ 280,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502153 + 0.147319i\)
\(L(\frac12)\) \(\approx\) \(0.502153 + 0.147319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.359i)T \)
5 \( 1 + (1.36 - 1.77i)T \)
7 \( 1 + (-1.45 - 2.21i)T \)
good3 \( 1 + (1.46 + 2.53i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.55 + 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.62iT - 13T^{2} \)
17 \( 1 + (-2.52 - 4.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.55 - 3.78i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.55 + 2.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.389iT - 29T^{2} \)
31 \( 1 + (-3.31 - 5.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.553 - 0.959i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.928iT - 41T^{2} \)
43 \( 1 + 3.21iT - 43T^{2} \)
47 \( 1 + (-2.82 - 1.63i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.44 - 9.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.71 - 3.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.21 - 5.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.80 - 1.61i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.22iT - 71T^{2} \)
73 \( 1 + (-1.58 - 2.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.81 + 3.93i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (9.69 + 5.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−11.96284164945483742551544683059, −11.07996259734680175058811276114, −10.35977012716001832279876771665, −8.656029464740599695604563301299, −7.909260646826595577468162437364, −7.22393875973799712116223626337, −6.16571931792750277658317159638, −5.53159676497915732855954823469, −2.82532917850268388539460003737, −1.34442485788640958299124779318, 0.68068263347489115113888585254, 3.37489614746412469519317960652, 4.61870705484139070018197579771, 5.37013638108617996077022996069, 7.22204770689937856578603552738, 7.974567169285678121585382374726, 9.409222803269403293139142484023, 9.765684132480635193962684138288, 10.79979032547006814085494085936, 11.50676904184883046207264445681

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