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Properties

Label 1-2280-2280.1883-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.545 + 0.838i$
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  + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)11-s + (0.984 + 0.173i)13-s + (0.342 − 0.939i)17-s + (−0.642 + 0.766i)23-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (0.173 + 0.984i)41-s + (0.642 + 0.766i)43-s + (0.342 + 0.939i)47-s + (0.5 + 0.866i)49-s + (0.642 − 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)11-s + (0.984 + 0.173i)13-s + (0.342 − 0.939i)17-s + (−0.642 + 0.766i)23-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (0.173 + 0.984i)41-s + (0.642 + 0.766i)43-s + (0.342 + 0.939i)47-s + (0.5 + 0.866i)49-s + (0.642 − 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.160101795 + 0.6292958654i\)
\(L(\frac12)\) \(\approx\) \(1.160101795 + 0.6292958654i\)
\(L(1)\) \(\approx\) \(0.9956452261 + 0.09098885284i\)
\(L(1)\) \(\approx\) \(0.9956452261 + 0.09098885284i\)

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 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.984 + 0.173i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.342 - 0.939i)T \)
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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

−19.38888444672591324019877136554, −18.79990677135939680669866542621, −18.3885355745237062746879056255, −17.240921066362623916539273326796, −16.628602172274956351261910096463, −16.01592773387011597592761148980, −15.26034699777368669061380213764, −14.5702436110664440516039214466, −13.57143785203125639879630357139, −13.147607882843319879182117468689, −12.236084396217636726032915380489, −11.61825295959402840972166054336, −10.67393635598094988485567002419, −10.10529745938148071299044520529, −9.00477950916239268281791506061, −8.68036708413790354377983165241, −7.73700769136868004312456967201, −6.69774051924847531620772649642, −5.8935129679122675906782036098, −5.67006352254931466628512805648, −4.06288671688172275710096004999, −3.66640147644028713243451273363, −2.69669225964284871141206009764, −1.69544991484859822123577424086, −0.5107629945649594865634022252, 0.99003060033032069671689061788, 1.90270245489162682163739385800, 3.095583733973593752836407431400, 3.765382779304822873503556927075, 4.532510383674061659164719013306, 5.61790922734865597471235033934, 6.34440508025492989024465655305, 7.18878579087541408794633547002, 7.67851227287230188788973850949, 8.961961360246028142953192045185, 9.43640314218644741666499676435, 10.15747167333442542565049681953, 11.02755274836920374759860521732, 11.7493121471082505739993628478, 12.6238486874423668571644592743, 13.191777303675559527058474030861, 14.02132431106976650468881664711, 14.60174760063896718250766093326, 15.64941823914432880348121542729, 16.19359913610983437048996540673, 16.74074723384366496402841521569, 17.80137831486285197644568455652, 18.17181340655986543618961866722, 19.20169148728235887106796802629, 19.75440668488638801413148820505

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