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Properties

Label 1-2015-2015.1668-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.615 - 0.788i$
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  + (0.866 − 0.5i)2-s + i·3-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s − 9-s + (0.5 − 0.866i)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 + (−0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)21-s − i·22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + i·3-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s − 9-s + (0.5 − 0.866i)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 + (−0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1668, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420022665 - 2.911476210i\)
\(L(\frac12)\) \(\approx\) \(1.420022665 - 2.911476210i\)
\(L(1)\) \(\approx\) \(1.681105043 - 0.5378960618i\)
\(L(1)\) \(\approx\) \(1.681105043 - 0.5378960618i\)

Euler product

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

−20.052347603546923413114860333940, −19.3896491980057648212194217227, −18.331940600828128536465830604430, −17.70991225396352354116996078968, −17.24886924976456228622124139918, −16.39072506017302190595748769499, −15.30483812652864743740345195369, −14.83212935083174300260325836450, −14.167575630039019035443334621982, −13.43831238642864272820174827085, −12.705409154666864792808598265410, −12.03042949157813833989700744295, −11.540193311701961914500424122599, −10.74148117266331934919813828019, −9.22786596214319552756412809203, −8.55828456628300893865043307390, −7.688902258462998645913410035585, −7.20444464554200239242363825029, −6.31509428597827196295924205594, −5.678173517268620818398344505412, −4.7641571558853900648055519178, −4.05992233931607995508760404824, −2.75119706279383690697959328519, −2.14673548508336876157657705273, −1.28321772459159122946617880037, 0.37432683264925759464954578030, 1.3958773746170162151098066900, 2.56524234092100643099277898185, 3.31175721340929924863467111045, 4.33104745738876407736399788640, 4.57263472674735059800886290087, 5.53564457888792490643870309127, 6.3435458293468872503713923030, 7.2184764476744051284885274030, 8.620645906966695178804416724464, 8.961948070349368743765730965129, 10.163243105839508933231376919089, 10.92025800290165541366090604994, 11.07705278060285389156960251907, 11.94904412652182637126379582097, 12.973369007940416368950636335512, 13.79952316105347801179506992144, 14.32419717789583678976832102537, 15.02844626112002101392690212562, 15.60006742827131251418001345512, 16.49357059462245258803593157305, 17.15990835118966205743950308526, 17.95918423690780211992506446747, 19.211316183015907828307602485854, 19.641336539667972297169892529035

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