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Properties

Label 1-2015-2015.583-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.398 - 0.917i$
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  + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + i·6-s − 7-s + 8-s + (0.5 − 0.866i)9-s − i·11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + i·17-s + (0.5 + 0.866i)18-s − i·19-s + (−0.866 + 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + i·6-s − 7-s + 8-s + (0.5 − 0.866i)9-s − i·11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + i·17-s + (0.5 + 0.866i)18-s − i·19-s + (−0.866 + 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.398 - 0.917i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.398 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021911761 - 0.6702792358i\)
\(L(\frac12)\) \(\approx\) \(1.021911761 - 0.6702792358i\)
\(L(1)\) \(\approx\) \(0.9510753001 + 0.02422011311i\)
\(L(1)\) \(\approx\) \(0.9510753001 + 0.02422011311i\)

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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 - T \)
11 \( 1 - iT \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + iT \)
43 \( 1 - iT \)
47 \( 1 - T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
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

−20.052522894871120497157575566525, −19.52149398278219609958202608448, −18.74125047857932584686551608578, −18.214143395952088500291817428651, −17.19596437807413628489147182890, −16.29815061169987943491057307890, −15.98086727405724698200501204790, −14.84105393561902729597384818132, −14.179778969586483828449659677734, −13.282790882409551813905903502707, −12.69711804637388538681733089237, −12.06495250528564217887064366942, −10.92285628835645054646464787647, −10.246478753885419822878420478984, −9.612698892449588994906340582943, −9.163776583845128217589287033311, −8.32248294729061569150844890470, −7.41271851610487156471377783232, −6.82800652973098969566727519126, −5.28700763641617200277614230674, −4.42153973785919223944163960551, −3.678875811381600528751169461280, −2.87732511576405261516437211484, −2.29266232551469299924933638213, −1.1647312564522277463493259905, 0.48916629819282077902881906645, 1.46262724943242813491936703584, 2.69893586663552081570638642686, 3.47327597185079984699312648238, 4.45391828385660688508660131272, 5.66792427484919958943883701642, 6.43392152030503785518168488781, 6.894349632116503042198254411779, 7.877146908356079915309308855902, 8.50006254042160285143974275251, 9.15663138446773545160040660115, 9.80426020773585010262147945370, 10.64143195003105540871745294667, 11.67094276084492351882528420149, 12.87679515854305312588271480191, 13.40525231327435290405100443409, 13.83082595088827102541964362728, 14.99944666485891548529006936836, 15.26058791792372109421482411193, 16.1569218775192522855351416165, 16.85753542218798148449645103888, 17.62197282652341047317903996754, 18.44985330888244079756721181762, 19.21612085031632627419094542000, 19.41139467611519366826066150062

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