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Properties

Label 2-280-56.19-c1-0-20
Degree $2$
Conductor $280$
Sign $-0.773 + 0.634i$
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  + (−1.40 + 0.183i)2-s + (−0.908 + 0.524i)3-s + (1.93 − 0.514i)4-s + (0.5 − 0.866i)5-s + (1.17 − 0.901i)6-s + (−2.14 − 1.54i)7-s + (−2.61 + 1.07i)8-s + (−0.949 + 1.64i)9-s + (−0.542 + 1.30i)10-s + (−1.17 − 2.03i)11-s + (−1.48 + 1.48i)12-s + 1.21·13-s + (3.29 + 1.77i)14-s + 1.04i·15-s + (3.47 − 1.98i)16-s + (−4.23 + 2.44i)17-s + ⋯
L(s)  = 1  + (−0.991 + 0.129i)2-s + (−0.524 + 0.302i)3-s + (0.966 − 0.257i)4-s + (0.223 − 0.387i)5-s + (0.480 − 0.368i)6-s + (−0.811 − 0.583i)7-s + (−0.924 + 0.380i)8-s + (−0.316 + 0.548i)9-s + (−0.171 + 0.413i)10-s + (−0.354 − 0.614i)11-s + (−0.428 + 0.427i)12-s + 0.336·13-s + (0.880 + 0.473i)14-s + 0.270i·15-s + (0.867 − 0.497i)16-s + (−1.02 + 0.593i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0633256 - 0.176985i\)
\(L(\frac12)\) \(\approx\) \(0.0633256 - 0.176985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.183i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.14 + 1.54i)T \)
good3 \( 1 + (0.908 - 0.524i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.17 + 2.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.21T + 13T^{2} \)
17 \( 1 + (4.23 - 2.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 + 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.59 + 4.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.21iT - 29T^{2} \)
31 \( 1 + (1.68 + 2.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.16 + 3.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.18iT - 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + (5.01 - 8.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.03 + 4.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.90 - 5.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.560 - 0.971i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.386 - 0.670i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.53iT - 71T^{2} \)
73 \( 1 + (-11.1 + 6.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.34 - 0.776i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 + (9.58 + 5.53i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.99830360687118193825230630952, −10.62579813115061503251463154483, −9.667439289453808609142177384262, −8.637301657121314060800308064630, −7.81678959290296928206710036294, −6.39428963192634129910889790589, −5.83606001397488189424325654060, −4.20308456559035650421699972474, −2.37273327250889114553447485825, −0.18881193094392034756360007709, 2.08230526102978484763373529619, 3.44726223608499805852609932288, 5.63773267672784554688636960148, 6.49381106133427862088895148293, 7.22451800971517407571899032264, 8.616245766500614296233031589718, 9.398938581538731986872251570207, 10.28519796410211664939472660241, 11.20389222273909238619696276495, 12.12988784648330216941649970859

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