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Properties

Label 1-2015-2015.522-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.279 + 0.960i$
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  + (−0.5 + 0.866i)2-s + i·3-s + (−0.5 − 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s − 9-s + (−0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s − i·22-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + i·3-s + (−0.5 − 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s − 9-s + (−0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.279 + 0.960i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5186769390 + 0.6908797346i\)
\(L(\frac12)\) \(\approx\) \(0.5186769390 + 0.6908797346i\)
\(L(1)\) \(\approx\) \(0.5694951667 + 0.3646282833i\)
\(L(1)\) \(\approx\) \(0.5694951667 + 0.3646282833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−19.2006357953289532822206208188, −18.862353710421072772751429136521, −18.36305099596076512239485954768, −17.49326910929025571448770505124, −17.10133377378935810105065823119, −16.03668309584692534935637236277, −15.052859006332499724019676036670, −14.24742317432800558735225054273, −13.25401675606342597578094555495, −12.86218867937981520031845840809, −12.21004787243786499401851215798, −11.14198685488639781517962145513, −11.098796368848313268038958302337, −9.88665730902293547723245291763, −8.81235131319636925657037517737, −8.39538593461332659733808026460, −7.820175800026429342812523132100, −6.787940631492729779819378214861, −5.861426486150412542014905681903, −5.006353221999193492140279441894, −3.92778080212381979337036118250, −2.71457550459572059847554271486, −2.315895173531219916140625231081, −1.46506818714178646120258602001, −0.358988053100173431153640664684, 0.45825904088564383894853060205, 1.76322843159375990684681016030, 2.959610649535771163093491388673, 4.21266540971264330187145764605, 4.750471418580444303043432328932, 5.313731431261908328258827248508, 6.4382833130907504020065320579, 7.22120987418340547005986832254, 7.97796977516783998490307850979, 8.76542854607903179667933507070, 9.48407211359678133039717571773, 10.2069760907172755379924413133, 10.93401611082369322925994498403, 11.31538614476716364499748841101, 12.980218860309488007693558711735, 13.50764707411818539132247783570, 14.47682744668103347398918751764, 15.01040470185580603395352851580, 15.58023489306710919902654561046, 16.42437534593809763410491258275, 16.88911584900038441278702138667, 17.778721573642409599133765776244, 18.06239515420641083352472849709, 19.35697052138610739049514155083, 19.91767677910879099113428563846

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