↑

Properties

Label 2-280-56.3-c1-0-27
Degree $2$
Conductor $280$
Sign $-0.553 + 0.832i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−0.275 − 0.158i)3-s − 2.00·4-s + (−0.5 − 0.866i)5-s + (0.224 − 0.389i)6-s + (−2.5 + 0.866i)7-s − 2.82i·8-s + (−1.44 − 2.51i)9-s + (1.22 − 0.707i)10-s + (−2.44 + 4.24i)11-s + (0.550 + 0.317i)12-s − 4.44·13-s + (−1.22 − 3.53i)14-s + 0.317i·15-s + 4.00·16-s + (−4.22 − 2.43i)17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.158 − 0.0917i)3-s − 1.00·4-s + (−0.223 − 0.387i)5-s + (0.0917 − 0.158i)6-s + (−0.944 + 0.327i)7-s − 1.00i·8-s + (−0.483 − 0.836i)9-s + (0.387 − 0.223i)10-s + (−0.738 + 1.27i)11-s + (0.158 + 0.0917i)12-s − 1.23·13-s + (−0.327 − 0.944i)14-s + 0.0820i·15-s + 1.00·16-s + (−1.02 − 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.832i$
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: \(1\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good3 \( 1 + (0.275 + 0.158i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.44 - 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + (4.22 + 2.43i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.67 + 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.94 + 2.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (0.775 - 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 - 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.02iT - 41T^{2} \)
43 \( 1 + 9.44T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.77 + 1.02i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.12 - 1.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.174 - 0.301i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.17 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (9.67 + 5.58i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.67 - 3.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.87iT - 83T^{2} \)
89 \( 1 + (-9.39 + 5.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.92390969901690724386654198877, −10.24814643986683716249685327374, −9.341104589415946978843128269344, −8.791461889856836695613114174551, −7.19732660168812778337160342315, −6.90435500159865840459380640733, −5.42183200585615445833258153681, −4.66926394236766159527419242616, −3.00268188933097995799999720595, 0, 2.53422944431611399476909669148, 3.46970536384755394067781227721, 4.90601835855517239878397692751, 5.98519094135951549798139863897, 7.53293967222389299818625715395, 8.482326698885140156043849878094, 9.680328219812060745610158771733, 10.43597722252966114039167660996, 11.18459593449970809714227803554

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