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Properties

Label 1-2280-2280.653-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.720 - 0.693i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  − i·7-s + 11-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + 31-s − i·37-s + (0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s − 49-s + (0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  − i·7-s + 11-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + 31-s − i·37-s + (0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s − 49-s + (0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.427443060 - 0.5754651308i\)
\(L(\frac12)\) \(\approx\) \(1.427443060 - 0.5754651308i\)
\(L(1)\) \(\approx\) \(1.066271680 - 0.1434850526i\)
\(L(1)\) \(\approx\) \(1.066271680 - 0.1434850526i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
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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

−19.60685326995681841026257374766, −19.199476482049919927388232150484, −18.25766591789829726204613914163, −17.52372382438612925539932252868, −17.091951077097472058031347551235, −15.965629038313576127218741872269, −15.486423946107479936534216323, −14.66978626812559252284490831569, −14.15156493024550797906449426960, −13.11948563957293936753562120032, −12.334356530835446192023113507195, −11.90527580919340822088731829780, −11.06223847326687043858522003034, −10.11339563986725149768690223178, −9.43311512017994203437327262763, −8.672094972328719340222948457539, −8.03582344766705855119983290500, −7.01271380228158784341695169650, −6.19472033054415383576516048274, −5.6251081583330785932254182217, −4.53137646746613580734841025137, −3.92984879105178537309506909354, −2.59808459509913022748116473197, −2.22925144209588323551591474839, −0.887021646580050001612430609186, 0.64626568632487247842773391376, 1.67430872138654735213151928929, 2.63267712280081402368463662706, 3.755335607160448357848008238900, 4.344084081450026466536653126477, 5.10359035830098171649958738478, 6.32724267862590043293463670571, 6.89615355491607261450542860009, 7.54233095293518614115922935336, 8.52990584065318319840891166101, 9.34821918940478544146063429016, 9.9917792905917087454348788001, 10.7730099280380433959510297389, 11.65737351116132665377606097058, 12.131864833507508063574051139118, 13.1630668785746997155648985454, 13.966520732110219864635525454536, 14.271827450610113930497130438585, 15.248055835972973096055514037613, 16.090398253591936912812391029209, 16.7507669863492100128374287531, 17.43927206768095949286833069312, 17.90418707331365453249726964525, 19.071454196373969351498360420365, 19.63841361420714492717280099273

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