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Properties

Label 2-280-280.69-c2-0-27
Degree $2$
Conductor $280$
Sign $0.812 - 0.583i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 1.18i)2-s + 3.23i·3-s + (1.18 + 3.81i)4-s + (−3.82 − 3.21i)5-s + (3.83 − 5.20i)6-s + (6.97 − 0.588i)7-s + (2.61 − 7.56i)8-s − 1.44·9-s + (2.34 + 9.72i)10-s + 13.5i·11-s + (−12.3 + 3.83i)12-s − 21.1i·13-s + (−11.9 − 7.32i)14-s + (10.3 − 12.3i)15-s + (−13.1 + 9.07i)16-s + 0.174·17-s + ⋯
L(s)  = 1  + (−0.805 − 0.592i)2-s + 1.07i·3-s + (0.296 + 0.954i)4-s + (−0.765 − 0.643i)5-s + (0.638 − 0.867i)6-s + (0.996 − 0.0840i)7-s + (0.327 − 0.945i)8-s − 0.160·9-s + (0.234 + 0.972i)10-s + 1.23i·11-s + (−1.02 + 0.319i)12-s − 1.63i·13-s + (−0.852 − 0.523i)14-s + (0.693 − 0.824i)15-s + (−0.823 + 0.567i)16-s + 0.0102·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.812 - 0.583i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ 0.812 - 0.583i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02032 + 0.328337i\)
\(L(\frac12)\) \(\approx\) \(1.02032 + 0.328337i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 1.18i)T \)
5 \( 1 + (3.82 + 3.21i)T \)
7 \( 1 + (-6.97 + 0.588i)T \)
good3 \( 1 - 3.23iT - 9T^{2} \)
11 \( 1 - 13.5iT - 121T^{2} \)
13 \( 1 + 21.1iT - 169T^{2} \)
17 \( 1 - 0.174T + 289T^{2} \)
19 \( 1 - 20.7T + 361T^{2} \)
23 \( 1 - 27.0iT - 529T^{2} \)
29 \( 1 - 18.8iT - 841T^{2} \)
31 \( 1 - 7.48iT - 961T^{2} \)
37 \( 1 - 66.6T + 1.36e3T^{2} \)
41 \( 1 - 40.1iT - 1.68e3T^{2} \)
43 \( 1 - 27.6T + 1.84e3T^{2} \)
47 \( 1 - 9.05T + 2.20e3T^{2} \)
53 \( 1 + 60.9T + 2.80e3T^{2} \)
59 \( 1 - 14.5T + 3.48e3T^{2} \)
61 \( 1 + 35.7T + 3.72e3T^{2} \)
67 \( 1 + 92.9T + 4.48e3T^{2} \)
71 \( 1 - 66.0T + 5.04e3T^{2} \)
73 \( 1 - 51.4T + 5.32e3T^{2} \)
79 \( 1 - 77.1T + 6.24e3T^{2} \)
83 \( 1 + 94.6iT - 6.88e3T^{2} \)
89 \( 1 + 59.6iT - 7.92e3T^{2} \)
97 \( 1 + 28.6T + 9.40e3T^{2} \)
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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

−11.47453943597100640990018044826, −10.72416634722972037412070951885, −9.821106872246246944395725863530, −9.141576213515054040794396117343, −7.83866986564890237963901486241, −7.55884132408657007094123675714, −5.18282337326576273989028791096, −4.38114302321231379948007801079, −3.24144909341283583641816954444, −1.24502868153281265494534473224, 0.871451901102943890856725661182, 2.32292168060215952233362182383, 4.40102534065926523485435240352, 6.02142751205550720666621698418, 6.84381318614773628604621869334, 7.70680218399374741182899581251, 8.281593425585146057185952809504, 9.372750072428485115546850335185, 10.84033639697842866574750697903, 11.42168586478016839017416922032

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