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Properties

Label 1-2015-2015.617-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.200 - 0.979i$
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 + (0.951 − 0.309i)3-s + (0.913 − 0.406i)4-s + (−0.866 + 0.5i)6-s + (0.104 − 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.809 − 0.587i)9-s + (0.406 + 0.913i)11-s + (0.743 − 0.669i)12-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (0.406 − 0.913i)17-s + (−0.669 + 0.743i)18-s + (−0.207 − 0.978i)19-s + (−0.207 − 0.978i)21-s + (−0.587 − 0.809i)22-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.951 − 0.309i)3-s + (0.913 − 0.406i)4-s + (−0.866 + 0.5i)6-s + (0.104 − 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.809 − 0.587i)9-s + (0.406 + 0.913i)11-s + (0.743 − 0.669i)12-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (0.406 − 0.913i)17-s + (−0.669 + 0.743i)18-s + (−0.207 − 0.978i)19-s + (−0.207 − 0.978i)21-s + (−0.587 − 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.200 - 0.979i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.200 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182367381 - 0.9652438218i\)
\(L(\frac12)\) \(\approx\) \(1.182367381 - 0.9652438218i\)
\(L(1)\) \(\approx\) \(1.003601754 - 0.2544235415i\)
\(L(1)\) \(\approx\) \(1.003601754 - 0.2544235415i\)

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 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (0.406 - 0.913i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 + (-0.406 + 0.913i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 - T \)
41 \( 1 + (0.743 - 0.669i)T \)
43 \( 1 + (-0.207 - 0.978i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (-0.743 - 0.669i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
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

−19.86399108048753982176314296035, −19.32298649106156683910471592099, −18.756251528964729185602588026081, −18.20743062536798668183213941376, −17.20124252276680814965488206969, −16.283512971119524574730163339567, −15.98794869570278710104112114393, −14.90308549721111286770216893611, −14.58103743428735328495227881921, −13.54904509619599536326009396910, −12.3909810152732280689604143435, −12.13557666429993615796159619634, −10.85122502944408252432177935307, −10.431320872327312825281279893780, −9.42253369690838734333439751461, −8.93497867627378465457428417836, −8.10939046212951817201657088154, −7.912748558412096138469297075990, −6.44338855296347994305386094425, −5.98940691661887173742134884971, −4.61758702256134532161602258165, −3.515773276556845421791053358701, −2.95338040230851556364757411687, −2.01914775384047944580545929747, −1.266846893426999388786638316503, 0.65800244219704105224651855146, 1.57673211690324112404270742196, 2.3779974660816482563003101137, 3.34411299552831618136843720407, 4.28029563099058759327647080573, 5.33922621923081928289550046242, 6.82256508725423523402182339021, 6.98238745005825647202818107542, 7.732709089010123550659576869433, 8.52347330158937025114491014573, 9.337483399569957071713895837451, 9.89720421397396638965474091106, 10.561395610256933305667468963413, 11.63977240327549190836651426161, 12.26943813460136675997986380830, 13.38887101804153139723684340124, 14.017376360672821661725195588674, 14.698157914727046842438410218780, 15.52410044133161540541926944676, 16.04678045201819669613384098644, 17.142740016911708958574819624712, 17.628528270326590042144936452102, 18.254049220101325813342405511392, 19.2366409172139229157205452874, 19.68942516553771759773501786279

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