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Properties

Label 1-1021-1021.275-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.897 - 0.441i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (−0.945 + 0.326i)2-s + (−0.982 + 0.183i)3-s + (0.786 − 0.617i)4-s + (0.869 − 0.494i)5-s + (0.869 − 0.494i)6-s + (−0.713 − 0.700i)7-s + (−0.542 + 0.840i)8-s + (0.932 − 0.361i)9-s + (−0.659 + 0.751i)10-s + (0.510 − 0.859i)11-s + (−0.659 + 0.751i)12-s + (−0.602 + 0.798i)13-s + (0.903 + 0.429i)14-s + (−0.763 + 0.645i)15-s + (0.237 − 0.971i)16-s + (−0.201 − 0.979i)17-s + ⋯
L(s)  = 1  + (−0.945 + 0.326i)2-s + (−0.982 + 0.183i)3-s + (0.786 − 0.617i)4-s + (0.869 − 0.494i)5-s + (0.869 − 0.494i)6-s + (−0.713 − 0.700i)7-s + (−0.542 + 0.840i)8-s + (0.932 − 0.361i)9-s + (−0.659 + 0.751i)10-s + (0.510 − 0.859i)11-s + (−0.659 + 0.751i)12-s + (−0.602 + 0.798i)13-s + (0.903 + 0.429i)14-s + (−0.763 + 0.645i)15-s + (0.237 − 0.971i)16-s + (−0.201 − 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.897 - 0.441i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7471851539 - 0.1737574478i\)
\(L(\frac12)\) \(\approx\) \(0.7471851539 - 0.1737574478i\)
\(L(1)\) \(\approx\) \(0.6248911620 + 0.02780832421i\)
\(L(1)\) \(\approx\) \(0.6248911620 + 0.02780832421i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.945 + 0.326i)T \)
3 \( 1 + (-0.982 + 0.183i)T \)
5 \( 1 + (0.869 - 0.494i)T \)
7 \( 1 + (-0.713 - 0.700i)T \)
11 \( 1 + (0.510 - 0.859i)T \)
13 \( 1 + (-0.602 + 0.798i)T \)
17 \( 1 + (-0.201 - 0.979i)T \)
19 \( 1 + (0.932 + 0.361i)T \)
23 \( 1 + (0.237 + 0.971i)T \)
29 \( 1 + (-0.713 + 0.700i)T \)
31 \( 1 + (0.631 + 0.775i)T \)
37 \( 1 + (0.687 - 0.726i)T \)
41 \( 1 + (0.932 + 0.361i)T \)
43 \( 1 + (0.989 + 0.147i)T \)
47 \( 1 + (0.903 + 0.429i)T \)
53 \( 1 + (0.997 + 0.0738i)T \)
59 \( 1 + (-0.478 - 0.878i)T \)
61 \( 1 + (0.903 + 0.429i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.542 + 0.840i)T \)
73 \( 1 + (0.903 - 0.429i)T \)
79 \( 1 + (-0.918 - 0.395i)T \)
83 \( 1 + (-0.982 + 0.183i)T \)
89 \( 1 + (-0.273 - 0.961i)T \)
97 \( 1 + (0.445 - 0.895i)T \)
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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

−21.77980986211945376113197188355, −20.89838769742330092191916198850, −19.937391887571315203822449890507, −19.061836539776577042016231112324, −18.44296264999069421136308865605, −17.72100300062272631137257551117, −17.21316713622929506751865552179, −16.55873344561914145506786947751, −15.42123870193383327158445309946, −14.97485537978219666975046183887, −13.3642153777323014774962741059, −12.63698514573461721968509552690, −12.082497179649417103187899766912, −11.13819230759418183432534931089, −10.26380593511564987463964697861, −9.79462014270191715411594700285, −9.063857480988040166759796947761, −7.71494964811739438932393836650, −6.92933755566250002012642524495, −6.191251329935924228041484629537, −5.55292364654348958282390501484, −4.13712517821729323284163040900, −2.71662346299835219825551510210, −2.11608072636332609490277417760, −0.8739226822078878031474054581, 0.73334538933498486065276557200, 1.437777134014428926948700846441, 2.884111953431155241549801394, 4.27996552437460806639248810045, 5.429937890017860451840460509891, 5.95721426840263544606898200423, 6.9149059593788157299632635282, 7.42390567174618653401456773027, 9.03615266414800349609300122635, 9.454890907803059260141032096071, 10.097649859006606106963668981487, 11.05343372402533757030994713153, 11.70975686059317250878481707981, 12.67385503291381156777551801346, 13.76977231927534650336311637187, 14.38054660063126239356838019939, 15.92423311376332699716347319946, 16.23553589746097116531614139287, 16.86889839281831716807229627421, 17.47772067048068254278398000904, 18.20191350947081965291128098752, 19.07560606928791742625834514432, 19.85166471617928049381080526765, 20.739910407214550343261264620128, 21.52579551325430834402942093319

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