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Properties

Label 1-2015-2015.1072-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.165 + 0.986i$
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  + (−0.978 + 0.207i)2-s + (−0.207 + 0.978i)3-s + (0.913 − 0.406i)4-s − i·6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.913 − 0.406i)9-s + (0.587 − 0.809i)11-s + (0.207 + 0.978i)12-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (0.587 + 0.809i)17-s + (0.978 + 0.207i)18-s + (0.951 + 0.309i)19-s + (−0.743 + 0.669i)21-s + (−0.406 + 0.913i)22-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (−0.207 + 0.978i)3-s + (0.913 − 0.406i)4-s − i·6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.913 − 0.406i)9-s + (0.587 − 0.809i)11-s + (0.207 + 0.978i)12-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (0.587 + 0.809i)17-s + (0.978 + 0.207i)18-s + (0.951 + 0.309i)19-s + (−0.743 + 0.669i)21-s + (−0.406 + 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1072, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8870020077 + 0.7502786182i\)
\(L(\frac12)\) \(\approx\) \(0.8870020077 + 0.7502786182i\)
\(L(1)\) \(\approx\) \(0.7268844035 + 0.3358151824i\)
\(L(1)\) \(\approx\) \(0.7268844035 + 0.3358151824i\)

Euler product

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

−19.84621528277658408797257734558, −18.830952893749120670972383527248, −18.298083607362379092979709081880, −17.66076528701328930702354029657, −17.17012362938620384628063178958, −16.4593325667255968551412297113, −15.600495152638528724823727424847, −14.42759073442129947632999812470, −14.071997309053318053761522638560, −12.934311371927112683993625722046, −12.13448358279202156602935160354, −11.66218587260764329695818116940, −10.97803719162156381079790227125, −10.06216165641352427042671211292, −9.35290848715538511387716166197, −8.27095967726722258824304284344, −7.858665452561397154084668578128, −6.98175258313883058356472982516, −6.62025211368738897819291523491, −5.39906380159150980938509577242, −4.45444462594622275423567561196, −3.15327220153537379093736452957, −2.30754797868167953882583132197, −1.36980396911579181267328414066, −0.80415672007111741177667444173, 0.89901000915647549785119657395, 1.8910056715744652287597370801, 3.0486205944140612613830896617, 3.81639543084033286060988248995, 5.062920668693146724850703387916, 5.733997883898790764728386680754, 6.291655236376156997914112647563, 7.59498807274432706143892326886, 8.29342562609105276793150861911, 8.9046782037434429179561390626, 9.63913875766518876757084259189, 10.33164893220760874197817843757, 11.15655127536928529045296734766, 11.66631229311515355262877589186, 12.277770473550802847352875069699, 13.93937310938200819720034032398, 14.46474677903975281954864087000, 15.190427155781079286414341947361, 15.88944366256981057237631058610, 16.44282659333949998459035912686, 17.26300496575422046239814612719, 17.72827880275979605818793126458, 18.55429175859862621880395784037, 19.37500329436439140644527982315, 20.01051081040632194258460291198

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