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Properties

Label 1-3021-3021.44-r1-0-0
Degree $1$
Conductor $3021$
Sign $-0.821 - 0.569i$
Analytic cond. $324.651$
Root an. cond. $324.651$
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  + (−0.909 + 0.416i)2-s + (0.653 − 0.757i)4-s + (0.452 + 0.891i)5-s + (0.428 − 0.903i)7-s + (−0.278 + 0.960i)8-s + (−0.783 − 0.621i)10-s + (0.0402 − 0.999i)11-s + (−0.872 − 0.488i)13-s + (−0.0134 + 0.999i)14-s + (−0.147 − 0.989i)16-s + (0.404 − 0.914i)17-s + (0.970 + 0.239i)20-s + (0.379 + 0.925i)22-s + (0.939 + 0.342i)23-s + (−0.589 + 0.807i)25-s + (0.996 + 0.0804i)26-s + ⋯
L(s)  = 1  + (−0.909 + 0.416i)2-s + (0.653 − 0.757i)4-s + (0.452 + 0.891i)5-s + (0.428 − 0.903i)7-s + (−0.278 + 0.960i)8-s + (−0.783 − 0.621i)10-s + (0.0402 − 0.999i)11-s + (−0.872 − 0.488i)13-s + (−0.0134 + 0.999i)14-s + (−0.147 − 0.989i)16-s + (0.404 − 0.914i)17-s + (0.970 + 0.239i)20-s + (0.379 + 0.925i)22-s + (0.939 + 0.342i)23-s + (−0.589 + 0.807i)25-s + (0.996 + 0.0804i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $-0.821 - 0.569i$
Analytic conductor: \(324.651\)
Root analytic conductor: \(324.651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (1:\ ),\ -0.821 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2703158913 - 0.8639648367i\)
\(L(\frac12)\) \(\approx\) \(0.2703158913 - 0.8639648367i\)
\(L(1)\) \(\approx\) \(0.7525754663 - 0.05964027401i\)
\(L(1)\) \(\approx\) \(0.7525754663 - 0.05964027401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.909 + 0.416i)T \)
5 \( 1 + (0.452 + 0.891i)T \)
7 \( 1 + (0.428 - 0.903i)T \)
11 \( 1 + (0.0402 - 0.999i)T \)
13 \( 1 + (-0.872 - 0.488i)T \)
17 \( 1 + (0.404 - 0.914i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.303 - 0.952i)T \)
31 \( 1 + (-0.0402 - 0.999i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + (-0.977 - 0.213i)T \)
43 \( 1 + (0.476 - 0.879i)T \)
47 \( 1 + (-0.998 + 0.0536i)T \)
59 \( 1 + (0.956 + 0.291i)T \)
61 \( 1 + (0.994 + 0.107i)T \)
67 \( 1 + (-0.982 - 0.186i)T \)
71 \( 1 + (-0.476 + 0.879i)T \)
73 \( 1 + (-0.589 - 0.807i)T \)
79 \( 1 + (0.964 - 0.265i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.897 - 0.440i)T \)
97 \( 1 + (0.998 + 0.0536i)T \)
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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

−19.28355967335092273351342211950, −18.37901093099335368722721887818, −17.66983275947325450557611158100, −17.29676035709614623777245825521, −16.54917617668568107130179395225, −15.9055478457607360267361033253, −14.958096368072507648528282299653, −14.5078199126390815258930779458, −13.17837718911050532090534579358, −12.53094287498964602711340215474, −12.16504982901854082368845029825, −11.46409301691557247357898341886, −10.37012957764861017651605552066, −9.89455102087426599703414621720, −9.04471365667619009174380629560, −8.66527955546303597266021245902, −7.89746845818104368988682012632, −6.98805455335668453404575715465, −6.26593508638144729118091048316, −5.062470620363350216244687395695, −4.72182409830534010786132805569, −3.4388946045658738499201428390, −2.44780437270236617656030811798, −1.74366445377468605729203757700, −1.20786651968270603608083148834, 0.22114236502562994682072153840, 0.87317577900769179955653794131, 1.99523093653322004962454546968, 2.799101512656017290506834861898, 3.613202928108163824337595217955, 4.97407679097829173544348656620, 5.59553165306578311187193951146, 6.451816744440477763266511951634, 7.27041977009926999399989598063, 7.568107721436213756848871350885, 8.49364696428999619478063515268, 9.39929277917333388570700823699, 10.087379064584071060219234399075, 10.56594129535567327848587557447, 11.33801728800634304800543976336, 11.81484872939051168407056019910, 13.298548224360280174826574513197, 13.832898529933480994803618025768, 14.53785630836381588089940994830, 15.04877724744570507456085820619, 15.9246218061087782073568789139, 16.71892143854430991884587559737, 17.30389793736248186078917742658, 17.745951858065658012539033127392, 18.590709075734176521418673410552

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