↑

Properties

Label 2-280-280.237-c1-0-36
Degree $2$
Conductor $280$
Sign $0.526 + 0.850i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 − 0.482i)2-s + (0.851 − 0.851i)3-s + (1.53 − 1.28i)4-s + (−2.18 + 0.451i)5-s + (0.721 − 1.54i)6-s + (2.50 − 0.844i)7-s + (1.42 − 2.44i)8-s + 1.55i·9-s + (−2.69 + 1.65i)10-s − 2.51i·11-s + (0.215 − 2.39i)12-s + (−2.05 + 2.05i)13-s + (2.92 − 2.33i)14-s + (−1.47 + 2.24i)15-s + (0.712 − 3.93i)16-s + (0.323 − 0.323i)17-s + ⋯
L(s)  = 1  + (0.940 − 0.340i)2-s + (0.491 − 0.491i)3-s + (0.767 − 0.641i)4-s + (−0.979 + 0.201i)5-s + (0.294 − 0.629i)6-s + (0.947 − 0.319i)7-s + (0.502 − 0.864i)8-s + 0.517i·9-s + (−0.851 + 0.523i)10-s − 0.757i·11-s + (0.0621 − 0.692i)12-s + (−0.570 + 0.570i)13-s + (0.782 − 0.623i)14-s + (−0.381 + 0.580i)15-s + (0.178 − 0.984i)16-s + (0.0784 − 0.0784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.526 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02198 - 1.12587i\)
\(L(\frac12)\) \(\approx\) \(2.02198 - 1.12587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.482i)T \)
5 \( 1 + (2.18 - 0.451i)T \)
7 \( 1 + (-2.50 + 0.844i)T \)
good3 \( 1 + (-0.851 + 0.851i)T - 3iT^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 + (2.05 - 2.05i)T - 13iT^{2} \)
17 \( 1 + (-0.323 + 0.323i)T - 17iT^{2} \)
19 \( 1 + 0.414iT - 19T^{2} \)
23 \( 1 + (5.60 - 5.60i)T - 23iT^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 - 7.18iT - 31T^{2} \)
37 \( 1 + (4.54 - 4.54i)T - 37iT^{2} \)
41 \( 1 - 1.40iT - 41T^{2} \)
43 \( 1 + (-4.10 - 4.10i)T + 43iT^{2} \)
47 \( 1 + (-5.84 + 5.84i)T - 47iT^{2} \)
53 \( 1 + (7.59 + 7.59i)T + 53iT^{2} \)
59 \( 1 + 1.24iT - 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (2.90 - 2.90i)T - 67iT^{2} \)
71 \( 1 + 0.972T + 71T^{2} \)
73 \( 1 + (10.4 + 10.4i)T + 73iT^{2} \)
79 \( 1 + 5.80iT - 79T^{2} \)
83 \( 1 + (9.17 - 9.17i)T - 83iT^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (-1.18 + 1.18i)T - 97iT^{2} \)
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

−11.66719589676986233024967111772, −11.18200676639252109502494682088, −10.17326912914481267381102025055, −8.536729775585251022574305011049, −7.65116782364547937386939851181, −6.94498338950817994885898592794, −5.37001668287632503007991962514, −4.33718206306083664361414902889, −3.18202514401719580773165184109, −1.72967171624585820970615118169, 2.48185055559919475888332037214, 3.94962231012931648655808188081, 4.54722710635059145223158215844, 5.78214209255928644375766870313, 7.26746039300027089412271285711, 7.989884715479448823050625678430, 8.882312957657471808525519063084, 10.27740547347753244166265048548, 11.36263569048776553431784237980, 12.24573549531947914235915411806

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