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Properties

Label 1-1021-1021.310-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.552 - 0.833i$
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  + (−0.478 − 0.878i)2-s + (−0.273 − 0.961i)3-s + (−0.542 + 0.840i)4-s + (−0.713 + 0.700i)5-s + (−0.713 + 0.700i)6-s + (−0.918 + 0.395i)7-s + (0.997 + 0.0738i)8-s + (−0.850 + 0.526i)9-s + (0.956 + 0.291i)10-s + (0.0184 − 0.999i)11-s + (0.956 + 0.291i)12-s + (−0.982 − 0.183i)13-s + (0.786 + 0.617i)14-s + (0.869 + 0.494i)15-s + (−0.412 − 0.911i)16-s + (−0.886 − 0.462i)17-s + ⋯
L(s)  = 1  + (−0.478 − 0.878i)2-s + (−0.273 − 0.961i)3-s + (−0.542 + 0.840i)4-s + (−0.713 + 0.700i)5-s + (−0.713 + 0.700i)6-s + (−0.918 + 0.395i)7-s + (0.997 + 0.0738i)8-s + (−0.850 + 0.526i)9-s + (0.956 + 0.291i)10-s + (0.0184 − 0.999i)11-s + (0.956 + 0.291i)12-s + (−0.982 − 0.183i)13-s + (0.786 + 0.617i)14-s + (0.869 + 0.494i)15-s + (−0.412 − 0.911i)16-s + (−0.886 − 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.552 - 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2630429949 - 0.1411345634i\)
\(L(\frac12)\) \(\approx\) \(0.2630429949 - 0.1411345634i\)
\(L(1)\) \(\approx\) \(0.3754922058 - 0.2195331354i\)
\(L(1)\) \(\approx\) \(0.3754922058 - 0.2195331354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.478 - 0.878i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.713 + 0.700i)T \)
7 \( 1 + (-0.918 + 0.395i)T \)
11 \( 1 + (0.0184 - 0.999i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (-0.886 - 0.462i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (-0.412 + 0.911i)T \)
29 \( 1 + (-0.918 - 0.395i)T \)
31 \( 1 + (0.237 + 0.971i)T \)
37 \( 1 + (-0.343 + 0.938i)T \)
41 \( 1 + (-0.850 - 0.526i)T \)
43 \( 1 + (0.975 + 0.219i)T \)
47 \( 1 + (0.786 + 0.617i)T \)
53 \( 1 + (-0.993 - 0.110i)T \)
59 \( 1 + (-0.999 - 0.0369i)T \)
61 \( 1 + (0.786 + 0.617i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.997 + 0.0738i)T \)
73 \( 1 + (0.786 - 0.617i)T \)
79 \( 1 + (0.572 - 0.819i)T \)
83 \( 1 + (-0.273 - 0.961i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (0.0922 + 0.995i)T \)
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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

−22.111453259989624980777385398897, −20.71367496219393417802785128166, −20.02495525278983426149628654570, −19.53423938634734999635521047292, −18.56207815834677429063199017114, −17.25061392453491781627433427953, −17.0016973767667693725396801350, −16.30535747802429407109508106375, −15.48101035005769896262406488847, −15.06231144606734128029449118667, −14.18683730269135344733281860268, −12.878916385528020009063472088355, −12.33851938361798530110674680139, −11.03943089753491521830180314439, −10.25235462556679486691297999992, −9.5299177429842857338266325248, −8.93119916880616901828869207779, −7.96490497971551571677526315803, −7.03337826407778737537601406126, −6.219328970846806319217499714702, −5.169564791175917853960977772372, −4.30910364453229231148233857702, −3.92800944349406496312738668133, −2.14246153749535114846907523006, −0.34461844398273455860760442382, 0.46915202309522580515026640105, 2.06019260862034654182316365662, 2.82624824032631821609779954079, 3.49358183466489804190017878432, 4.81328724376071179890745163631, 6.10451477386819276392262426573, 6.95182812255633406369678880315, 7.66615194494800371650972079595, 8.564047478219908578546245050495, 9.356203297860950499265653367158, 10.52133693633918013264680234277, 11.16487161072872733540373943356, 11.907722858667852299031627514048, 12.4896305285176017807275253098, 13.386757385844531970729392930476, 13.98746359148152146379775730126, 15.27634774251676787307146547510, 16.14512775255296128858555466028, 17.08893446833454600208146121911, 17.774924931208350648149102229724, 18.65650487045087916854867637524, 19.26363112043691380697052713587, 19.47284932360300650943049194611, 20.34562935408818959096319850165, 21.80189774014949484508979949779

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