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Properties

Label 1-2015-2015.1083-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.999 + 0.0400i$
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  + (0.743 + 0.669i)2-s + (0.207 + 0.978i)3-s + (0.104 + 0.994i)4-s + (−0.5 + 0.866i)6-s + (0.994 − 0.104i)7-s + (−0.587 + 0.809i)8-s + (−0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)14-s + (−0.978 + 0.207i)16-s + (0.406 + 0.913i)17-s + (−0.951 − 0.309i)18-s + (−0.978 − 0.207i)19-s + (0.309 + 0.951i)21-s + (0.406 + 0.913i)22-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.207 + 0.978i)3-s + (0.104 + 0.994i)4-s + (−0.5 + 0.866i)6-s + (0.994 − 0.104i)7-s + (−0.587 + 0.809i)8-s + (−0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)14-s + (−0.978 + 0.207i)16-s + (0.406 + 0.913i)17-s + (−0.951 − 0.309i)18-s + (−0.978 − 0.207i)19-s + (0.309 + 0.951i)21-s + (0.406 + 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.999 + 0.0400i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.999 + 0.0400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05822416192 + 2.905850795i\)
\(L(\frac12)\) \(\approx\) \(0.05822416192 + 2.905850795i\)
\(L(1)\) \(\approx\) \(1.123608294 + 1.433225077i\)
\(L(1)\) \(\approx\) \(1.123608294 + 1.433225077i\)

Euler product

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

−19.50572833884685192914653842516, −18.99433674000155479894034648616, −18.40690801147101332269510789144, −17.52874465029006607855225662840, −16.87750826668441420719084439669, −15.67745593538706715994553877804, −14.74982545857951072125642128349, −14.26545390176948213086921816387, −13.81135728355969246236607164029, −12.869667257481001881622357674970, −12.209367774846514940746479758809, −11.59566367484826420203011507523, −11.039153996916188017675251441149, −10.107673428421363308685627050434, −8.80502846356398901042535822544, −8.63580736205209498318415069571, −7.19898142557390379642346903723, −6.77930995384117433735438917913, −5.69699893777872127536792395219, −5.11307849657955255381890687454, −4.05804530349014751370716332999, −3.19489725620896944432435801276, −2.2890042763125388153353121251, −1.54016252754755083097042643973, −0.74466091777236556991691175499, 1.57166267546324875652061040447, 2.66608931788363341456368710570, 3.60961414728765145846213742879, 4.41200335800450543681574758306, 4.76357994187182793819324381774, 5.76106683425877569374317637882, 6.525658725389379024661507698754, 7.51345101065127320989457212930, 8.41503483252526878220293150548, 8.776539553270483749649213264627, 9.86543535669193656744979106398, 10.79756550102744574411070559508, 11.47183913432427543122022602615, 12.18144434264786083649092004373, 13.13753504289783574398305425103, 14.026388581836154846147372998521, 14.61979964433869795779919827440, 15.06632492884611514715671765966, 15.652900037819269593060654964938, 16.73570641621170249516322750443, 17.19942802991925907120312649310, 17.566251658586926532742388352996, 18.92824794817792264121000249876, 19.82042163157358300147335402716, 20.579786125764548226263846922709

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