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Properties

Label 1-2015-2015.719-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.254 + 0.966i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s + 22-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s + 22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.254 + 0.966i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.254 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09243371934 + 0.1199609730i\)
\(L(\frac12)\) \(\approx\) \(0.09243371934 + 0.1199609730i\)
\(L(1)\) \(\approx\) \(0.8460181956 - 0.3559784896i\)
\(L(1)\) \(\approx\) \(0.8460181956 - 0.3559784896i\)

Euler product

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

−19.4197282926474727227827417674, −18.88168855272876574323224284496, −18.11505723451849423224811087034, −17.49791182668597102337175310748, −16.35605079364885049742311253617, −15.74708963799199805483209425000, −15.36226659222527081790763351623, −14.63294653615010023194110914638, −13.550573137530474608293562860339, −13.45434452677842466889047427119, −12.369830145319842511258421713669, −11.214911834209778047477245283786, −10.313775460618928424900377876213, −9.52098073052876400562258212483, −8.97932511835320184408287739591, −8.27048327641765439449281548303, −7.73012895876902100724190670861, −6.70404441193446508674747867687, −6.02965774396856622540602503400, −5.186455486414311671384589995, −4.17859021774398369749829694316, −3.164411727711816830288143306094, −2.32353648685356576471699451573, −1.28271741773471869662852141918, −0.028375666385739780530252666856, 1.0568889276026300882809950105, 2.03494809884875795086691029394, 2.7504880603758551347827547641, 3.625101298830388761954938672865, 4.24310552078013435798534232448, 5.124481105425026226102440080844, 6.87257041337418550185703311871, 7.27040301439047599143188770606, 8.10213554899080589948484026458, 8.86976565158667897355258103780, 9.65156464703407052081786309945, 10.24716496487780765289949372545, 10.70937130642416101349450647283, 12.04849315165583994871757196313, 12.54024034663505115065872658766, 13.362427194623808839043860862727, 13.91732098048575642169478623735, 14.59851414295401319254474693596, 15.88728704574760249019878793439, 16.17636926122972844706784316039, 17.27943496619928553532594928900, 18.023910405184094782141206856649, 18.651857684956911098522323520944, 19.34749344679157026715297793917, 20.13410181512038453056900628338

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