Properties

Label 1-280-280.227-r1-0-0
Degree $1$
Conductor $280$
Sign $-0.777 - 0.629i$
Analytic cond. $30.0901$
Root an. cond. $30.0901$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s i·13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s i·27-s + 29-s + (−0.5 − 0.866i)31-s + (−0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s − 41-s i·43-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s i·13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s i·27-s + 29-s + (−0.5 − 0.866i)31-s + (−0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s − 41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.777 - 0.629i$
Analytic conductor: \(30.0901\)
Root analytic conductor: \(30.0901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (1:\ ),\ -0.777 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5508495101 - 1.556425609i\)
\(L(\frac12)\) \(\approx\) \(0.5508495101 - 1.556425609i\)
\(L(1)\) \(\approx\) \(1.114803552 - 0.4822781678i\)
\(L(1)\) \(\approx\) \(1.114803552 - 0.4822781678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.89917375519856039506759673317, −25.00996890542416849444676354006, −24.0137149821975039738835099450, −23.066373575739304031488055773751, −21.780978405764138119687601068245, −21.34783958047517607448391250337, −20.10745754674758980486028121299, −19.71004023614532179386814111700, −18.486142283184810406911468975010, −17.58137834338805284294934430148, −16.24756962527978145933799492472, −15.5957251786539563644927117889, −14.64810189373547782526120965049, −13.75274920238567865553730674935, −12.89153729214914493550191478458, −11.59342411027073447611029651454, −10.482517570697316569290367472, −9.51937307388044753325978146623, −8.7466921908754274286855559452, −7.598362621421491656962540331, −6.627822889087063794229710926533, −4.896197741510242668892740344103, −4.22411058318721066964624129525, −2.79362012237361480003593515889, −1.84011330626874518932297031160, 0.41436846916003780654128931825, 1.95290919458244534062908280059, 3.04448685912404642789711656076, 4.11955803711179041878400958572, 5.726541193141967032007712903721, 6.71141362491531381714580636208, 8.155961717035091958308996808785, 8.36742116209516728560209099308, 9.83239793709113105070853860732, 10.75911488180791702285367403323, 12.125809277393788702301251235251, 13.05565161897819355637681184976, 13.74687655495938705144354846040, 14.83409188188457317045697245964, 15.58185249301214445807606308824, 16.75299564713503196090648713781, 18.00173824363871870950842279796, 18.62216065304109617473401852968, 19.652336875806843959891913873907, 20.348011394557313957794043427106, 21.285295201946106265241957663242, 22.25410235566173433845720942143, 23.4934040720527903580907011659, 24.21615602064077802178779652070, 25.05519507354105617364879193958

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