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Properties

Label 2-280-56.3-c1-0-26
Degree $2$
Conductor $280$
Sign $0.841 + 0.539i$
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  + (1.06 − 0.926i)2-s + (1.90 + 1.10i)3-s + (0.281 − 1.98i)4-s + (0.5 + 0.866i)5-s + (3.05 − 0.591i)6-s + (0.584 − 2.58i)7-s + (−1.53 − 2.37i)8-s + (0.922 + 1.59i)9-s + (1.33 + 0.461i)10-s + (−2.90 + 5.03i)11-s + (2.71 − 3.46i)12-s − 4.83·13-s + (−1.76 − 3.29i)14-s + 2.20i·15-s + (−3.84 − 1.11i)16-s + (3.78 + 2.18i)17-s + ⋯
L(s)  = 1  + (0.755 − 0.655i)2-s + (1.10 + 0.635i)3-s + (0.140 − 0.990i)4-s + (0.223 + 0.387i)5-s + (1.24 − 0.241i)6-s + (0.220 − 0.975i)7-s + (−0.542 − 0.840i)8-s + (0.307 + 0.532i)9-s + (0.422 + 0.145i)10-s + (−0.875 + 1.51i)11-s + (0.784 − 0.999i)12-s − 1.34·13-s + (−0.472 − 0.881i)14-s + 0.568i·15-s + (−0.960 − 0.279i)16-s + (0.917 + 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.841 + 0.539i$
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: \((2,\ 280,\ (\ :1/2),\ 0.841 + 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38221 - 0.697861i\)
\(L(\frac12)\) \(\approx\) \(2.38221 - 0.697861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.926i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.584 + 2.58i)T \)
good3 \( 1 + (-1.90 - 1.10i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.90 - 5.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.83T + 13T^{2} \)
17 \( 1 + (-3.78 - 2.18i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.63 + 0.945i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.157 - 0.0911i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.38iT - 29T^{2} \)
31 \( 1 + (-2.03 + 3.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.69 - 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.44iT - 41T^{2} \)
43 \( 1 + 2.10T + 43T^{2} \)
47 \( 1 + (0.946 + 1.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.54 - 4.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.35 + 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.89 - 8.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.04 + 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + (-7.34 - 4.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.9iT - 83T^{2} \)
89 \( 1 + (5.11 - 2.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−11.92043632591920973582219277036, −10.43188897980570000264729325156, −10.13479660558828603257620273513, −9.396935833327469244764006397054, −7.79115980727282689514901257460, −6.98198274783155421432613518338, −5.20313561279848941475356847784, −4.32889459366383985420056812152, −3.20903686494542219535484784038, −2.12344675773668501207039602254, 2.42777943088373480441538426415, 3.19201072659586267901510387329, 5.08785115953684973434899949826, 5.74593730723812677711729608909, 7.21184010463788570796754248618, 8.141370767905101533845288673105, 8.552157048394153547716652737748, 9.722882615946267437858658845273, 11.43724857424098103533683128087, 12.31115218171866277483953697236

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