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Properties

Label 1-2015-2015.114-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.595 - 0.803i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + (−0.978 − 0.207i)2-s + (0.309 − 0.951i)3-s + (0.913 + 0.406i)4-s + (−0.5 + 0.866i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.913 + 0.406i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + (0.913 − 0.406i)17-s + (0.669 + 0.743i)18-s + (0.978 + 0.207i)19-s + (−0.978 − 0.207i)21-s + (−0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.309 − 0.951i)3-s + (0.913 + 0.406i)4-s + (−0.5 + 0.866i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.913 + 0.406i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + (0.913 − 0.406i)17-s + (0.669 + 0.743i)18-s + (0.978 + 0.207i)19-s + (−0.978 − 0.207i)21-s + (−0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.595 - 0.803i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.595 - 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8375116712 - 1.663996749i\)
\(L(\frac12)\) \(\approx\) \(0.8375116712 - 1.663996749i\)
\(L(1)\) \(\approx\) \(0.7688692810 - 0.4820570557i\)
\(L(1)\) \(\approx\) \(0.7688692810 - 0.4820570557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.978 - 0.207i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−19.78148163694650638836566526082, −19.31397324140701645014498648101, −18.6797265369862373355105728957, −17.73679035009111871579868685637, −16.99120533339702894598892575944, −16.43086544427450071984781999188, −15.63297834052177354923800435757, −15.19520325080828083461652165565, −14.42424111092765574413703658382, −13.73417017158091966555673919761, −12.23330977472117240494475038876, −11.74785357419119340169194439748, −10.96465799655630533594091624721, −10.119575633929155080078907671955, −9.47163903754278727391879877636, −8.8630555334231464732961251019, −8.345923495045360465769551507485, −7.38163441211975094380162033021, −6.350209787520675093421614696730, −5.60058721215431015002314538666, −4.92744317675068328398494565929, −3.47001004356656531044589825867, −3.01066722715009870008024365812, −1.90449403970227876351280492447, −0.85621754988679836501457553270, 0.58785722791315356426530033525, 1.128629941850371829485006887204, 1.922008580309504575962755093974, 3.13814698414431473486762094663, 3.5788772668553086342180724677, 5.028248020474384053535826021047, 6.28004979738718876459696268869, 6.98872777117700860699618313216, 7.34125600475737151338247881509, 8.23926122648403488164314882881, 8.94267957915357004512901917193, 9.81038356962962932533244177358, 10.39124514923283598591408428044, 11.534740542175331396134128490914, 11.93581767284432396146263308902, 12.72651623288342102879818863957, 13.62674048869168352773745502954, 14.31282211307945205196300425768, 15.06342300327905496907942824126, 16.17878615364871015994632882575, 16.915520035914857991163408336937, 17.30681269958197382845535225236, 18.19970533092008087201164841804, 18.7011544441435233696661428668, 19.56603583029399916684226242616

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