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Properties

Label 1-3021-3021.44-r1-0-0
Degree 11
Conductor 30213021
Sign −0.821−0.569i-0.821 - 0.569i
Analytic cond. 324.651324.651
Root an. cond. 324.651324.651
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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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

Λ(s)=(3021s/2 ΓR(s+1) L(s)=((−0.821−0.569i) Λ‾(1−s)\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}
Λ(s)=(3021s/2 ΓR(s+1) L(s)=((−0.821−0.569i) Λ‾(1−s)\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: 11
Conductor: 30213021    =    3⋅19⋅533 \cdot 19 \cdot 53
Sign: −0.821−0.569i-0.821 - 0.569i
Analytic conductor: 324.651324.651
Root analytic conductor: 324.651324.651
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3021(44,⋅)\chi_{3021} (44, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 3021, (1: ), −0.821−0.569i)(1,\ 3021,\ (1:\ ),\ -0.821 - 0.569i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.2703158913−0.8639648367i0.2703158913 - 0.8639648367i
L(12)L(\frac12) ≈\approx 0.2703158913−0.8639648367i0.2703158913 - 0.8639648367i
L(1)L(1) ≈\approx 0.7525754663−0.05964027401i0.7525754663 - 0.05964027401i
L(1)L(1) ≈\approx 0.7525754663−0.05964027401i0.7525754663 - 0.05964027401i

Euler product

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