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Properties

Label 1-2015-2015.1172-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.800 + 0.598i$
Analytic cond. $9.35762$
Root an. cond. $9.35762$
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 + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s − 7-s + 8-s + (0.5 − 0.866i)9-s + i·11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − i·17-s + (0.5 + 0.866i)18-s + i·19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s − 7-s + 8-s + (0.5 − 0.866i)9-s + i·11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − i·17-s + (0.5 + 0.866i)18-s + i·19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.800 + 0.598i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.800 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1486783636 + 0.4472377421i\)
\(L(\frac12)\) \(\approx\) \(0.1486783636 + 0.4472377421i\)
\(L(1)\) \(\approx\) \(0.4359692267 + 0.2671318837i\)
\(L(1)\) \(\approx\) \(0.4359692267 + 0.2671318837i\)

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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 - T \)
11 \( 1 + iT \)
17 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 + iT \)
47 \( 1 - T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (0.866 + 0.5i)T \)
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

−19.60052880962328725233614525882, −18.983998453694930998470708747747, −18.236826610512867210833398441006, −17.63959264514679999805406784943, −16.83944632784017324641283774699, −16.29729692282124567398944866368, −15.654898897596856956771701072895, −14.19905242322095106942555730689, −13.3153513055308682760352862504, −12.96734333097524060172545277169, −12.18339683097865823958486671440, −11.47157978961775615869964719254, −10.78185525118070508900712853881, −10.19353184538752739426571695447, −9.316055400277991861743966790008, −8.486327964831475162396975845764, −7.68033836957097963314661744835, −6.74835365831528161695564414279, −6.0722941818116599101620993484, −5.13590012552373161025461139530, −4.05796332022893575504785748069, −3.26013670149714495237081400149, −2.328082543197869872926827004001, −1.29262424513654896942655542258, −0.3467530686221838045490547471, 0.73338171604257906678064512881, 2.02704316675604585033345481726, 3.45696037625400323002064745453, 4.44713900968784499242308607668, 5.01248997898410039982943726046, 6.11357029358869241003933516988, 6.41196964574802294529880014622, 7.298473070704755956225643748257, 8.11732598800844735203150696712, 9.301585580289138260241331092460, 9.84083275226685394303525409656, 10.16462203048437685904279059149, 11.21429834869546391712763923646, 12.17195578616460001138098263339, 12.77614131174278496534273742722, 13.817578122455515214915573923768, 14.59669479260795765315455229285, 15.51245371833206665560106761130, 15.8851724268390174905582813478, 16.59344559347917790808132910186, 17.14292262928352704724227456216, 17.99224784615144910737850423811, 18.468562232752104719038814261988, 19.28482989672254754040214663106, 20.22730777747922406786534866481

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