Properties

Label 1-280-280.243-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

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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: $\chi_{280} (243, \cdot )$
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 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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