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Properties

Label 1-1015-1015.202-r0-0-0
Degree $1$
Conductor $1015$
Sign $0.525 - 0.850i$
Analytic cond. $4.71364$
Root an. cond. $4.71364$
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  − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s − 11-s + i·12-s + i·13-s + 16-s + i·17-s + i·18-s − 19-s + i·22-s − i·23-s + 24-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s − 11-s + i·12-s + i·13-s + 16-s + i·17-s + i·18-s − 19-s + i·22-s − i·23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1015\)    =    \(5 \cdot 7 \cdot 29\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(4.71364\)
Root analytic conductor: \(4.71364\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1015} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1015,\ (0:\ ),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7893996699 - 0.4401191415i\)
\(L(\frac12)\) \(\approx\) \(0.7893996699 - 0.4401191415i\)
\(L(1)\) \(\approx\) \(0.6617987563 - 0.4901941228i\)
\(L(1)\) \(\approx\) \(0.6617987563 - 0.4901941228i\)

Euler product

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

−21.82312270229038325098672238246, −21.12155244239185611058401169742, −20.352130988457197948209956530019, −19.33653409188774657741983120653, −18.382902735584542668943075330012, −17.58226858278770227837069020792, −16.970254340832334515442765715131, −16.040796211726501266151174394926, −15.46291938717321707814669580790, −15.04214064192111604362649739676, −13.94786356283130053255968673301, −13.343637835344611674936379517714, −12.3335695053067711312366235807, −11.14314693318393228039738754884, −10.21818042481765528009723027441, −9.70709496509257508587564877482, −8.635623623379755183261526098468, −8.056083367345658165251909417193, −7.09699812708627397530798850557, −5.91346061175355082983153279693, −5.2748314604914725205454466223, −4.57683885132616220884987342120, −3.53741331405148931931332023380, −2.61346308486070065576045482854, −0.49901326454250295795399057453, 0.90236246237444444838549540776, 2.11036298731481838767081551925, 2.536597652512953362270693713194, 3.84145965634698139890705608392, 4.76801308236269482454618277637, 5.87803099805988477549518088457, 6.7060409974495331402589346032, 7.98134221751357215316203346037, 8.436973401305200677851067343964, 9.41801482876526472253984827707, 10.536571344383343234396699422116, 11.09958162152062798972129437729, 12.03670855608964686932570755367, 12.81090685711075339891211637847, 13.16864564186653137611331069653, 14.24360973387817197857553014081, 14.73799549408889094013565512028, 16.180393078370830736278844074729, 17.1518554542438585142878610356, 17.80778725257730295608294284089, 18.56461447465240711900469984670, 19.28643288870695925583589733765, 19.59739389386546247533189517628, 20.95355513208364414726079563964, 21.10432453846603928142181735359

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