Properties

Label 1-280-280.187-r1-0-0
Degree $1$
Conductor $280$
Sign $-0.910 - 0.413i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.910 - 0.413i$
Analytic conductor: \(30.0901\)
Root analytic conductor: \(30.0901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (1:\ ),\ -0.910 - 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1068181501 - 0.4929036306i\)
\(L(\frac12)\) \(\approx\) \(0.1068181501 - 0.4929036306i\)
\(L(1)\) \(\approx\) \(0.6889802340 - 0.1544277387i\)
\(L(1)\) \(\approx\) \(0.6889802340 - 0.1544277387i\)

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

−25.92673447921288610076438590307, −24.78851097235565389032480656443, −23.65901337433049901212115376986, −23.24091886378836211168504048658, −22.09128647859151496114972743554, −21.27464326202443012929432107890, −20.735245420603494226914256527934, −19.111802267500738684738498589587, −18.55842627233528209845673727892, −17.33760040622384528729245435303, −16.53648209279976973836056998845, −15.91235754018163342072143911835, −14.74879435475674686246107132918, −13.728334901940497464928112315124, −12.496314376324470876069382302825, −11.61236789118757472385037007891, −10.77545002329182342950367723893, −9.8178988609538678155497607022, −8.78884588164817186881605422227, −7.43402913082105408129150908659, −6.25003374250426426870863217246, −5.395541311452557840241878863720, −4.29298816798447801553070571009, −3.12251848664993946698000014927, −1.28732340565882654189743057539, 0.18787731942642221200142912318, 1.55596407272335668305734840413, 2.96445274674087308301386672460, 4.68230888320562073635912097428, 5.44547079047866766427347533083, 6.66612125936894954433205600596, 7.506757201022014323194243630755, 8.61664832417528990021509443504, 10.244107815097218567275203650309, 10.675106069613032795050140914592, 12.07755377529366530985956405274, 12.66124499867112159387512438220, 13.56802843319872056393774307950, 14.96664517279997765444199981378, 15.77642531159776357008610348211, 16.98996453238912716048103359320, 17.593178290994521442661262891222, 18.4737846503174806024846923588, 19.391402977692269044199189176465, 20.45948661170048461559568185666, 21.51851744568476551432055680786, 22.456751745991947795057077670076, 23.2945871945280924681716452581, 23.83476277222804275890421225028, 25.090405249392831509467126580626

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