Properties

Label 1-2015-2015.393-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.789 + 0.613i$
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  + (−0.207 + 0.978i)2-s + (−0.951 − 0.309i)3-s + (−0.913 − 0.406i)4-s + (0.5 − 0.866i)6-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.913 − 0.406i)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 + (−0.743 + 0.669i)18-s + (0.978 + 0.207i)19-s + (0.978 + 0.207i)21-s + (0.587 − 0.809i)22-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.951 − 0.309i)3-s + (−0.913 − 0.406i)4-s + (0.5 − 0.866i)6-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.913 − 0.406i)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 + (−0.743 + 0.669i)18-s + (0.978 + 0.207i)19-s + (0.978 + 0.207i)21-s + (0.587 − 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4765019922 + 0.1632270890i\)
\(L(\frac12)\) \(\approx\) \(0.4765019922 + 0.1632270890i\)
\(L(1)\) \(\approx\) \(0.4968549657 + 0.1740329804i\)
\(L(1)\) \(\approx\) \(0.4968549657 + 0.1740329804i\)

Euler product

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

−20.01580277176309366832053695223, −19.01750789788908947015790913592, −18.40117321965867458948644618611, −17.84886489689695361135176847829, −17.11099052688283955882617811267, −16.13031454710031348874963486481, −15.92947517178150108547057473975, −14.73127615533396685466290976915, −13.57673807090555421318381944029, −13.102634480110646549797316172143, −12.29777526788842895470460438462, −11.747002097354066375821745395006, −10.93355337749021634793722063207, −10.25471155483807716011336459349, −9.557813315851955466647772592838, −9.20742738984891726156495747176, −7.64670927291108458655917485443, −7.24365006816564515268972752635, −5.89271324310134121052302920871, −5.294660615324987781318977251312, −4.46797270346092903100518182228, −3.50590273895532336801274850613, −2.86479265623942538010889424074, −1.6409942839646239867769947403, −0.51704285107465739570131928068, 0.45050856518595484145525415991, 1.616840136355241208981591597574, 3.06917709537190799360916373409, 4.0897437277330288046525985172, 5.06726993629718716631265146019, 5.821970599594518416601139372, 6.22261109780246325392496537684, 7.1179233488292151987739356321, 7.81863239792736727605432184360, 8.57593591357580339180614165767, 9.77427269905987184546775290535, 10.12987062859526509860815458767, 10.9963603431286706336270285833, 12.05453654723683004566187552274, 12.95361697145375219483848576813, 13.18727113414054569738974757776, 14.196948946609956300172872526227, 15.14886631033445640159637112910, 15.94257170306656021573496285031, 16.36559918472653242019222491750, 16.9199171856454529843449021952, 17.818550270873785490512946035775, 18.45730110471401756619635056566, 18.93801046051180449668237512650, 19.64267658955376066776077934278

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