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Properties

Label 1-2015-2015.99-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.840 + 0.541i$
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  − i·2-s + (−0.5 + 0.866i)3-s − 4-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)21-s + (−0.5 − 0.866i)22-s + ⋯
L(s)  = 1  − i·2-s + (−0.5 + 0.866i)3-s − 4-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)21-s + (−0.5 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.840 + 0.541i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.840 + 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07835536671 - 0.2661382248i\)
\(L(\frac12)\) \(\approx\) \(-0.07835536671 - 0.2661382248i\)
\(L(1)\) \(\approx\) \(0.5799166886 - 0.2664587144i\)
\(L(1)\) \(\approx\) \(0.5799166886 - 0.2664587144i\)

Euler product

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

−19.99695697254982200453023779148, −19.35010301857459804464902426178, −18.803091540317716737310826660187, −18.07504082012840062272209339036, −17.40706026169174780773340618426, −16.622703500712700388105699485569, −16.33338244033926039519270251135, −15.171629630774975362842475781551, −14.62343688156691053231499256519, −13.83064228710504407025843504694, −12.83393592970491657243945012825, −12.610138509279470575979253620300, −11.815467344658793011506907660563, −10.65614500470058577804012161729, −9.74772952665259117855771183470, −9.05506440494637196769525129800, −8.1168450270729638172994228431, −7.531068180856798159093630920705, −6.54672503171903720976796113708, −6.13832751447705872609957009009, −5.58036063442282127785399761785, −4.374142871336967243647324470752, −3.62169382794755377151172531233, −2.27131596522726065702702779201, −1.261381827440633204280543181184, 0.12605977708911242524091227440, 1.052983742485913411960474081415, 2.46829996304881615242051836066, 3.378462636846217873999523905291, 3.97099523627763209766391040270, 4.62576990169631064568053832517, 5.721432816367618450346173651330, 6.31615830469873047226491564946, 7.49246939072596533573622296057, 8.70778872780234174652052517103, 9.36196031665191313189225484140, 9.84915149963520548340690998663, 10.66722114884480852358690062683, 11.26359861068004706073021537573, 11.98162272528070397451168671807, 12.68120152219995158511229844685, 13.59457760359746654273962230953, 14.25926850054726764158837306043, 15.02898660941808498917412203743, 16.12257845662930518485140461674, 16.62664337475806754834482185716, 17.31920208972802779179641152584, 18.04226433311599557272199817816, 19.055745188380557080666324530917, 19.5275821268379537914836526027

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