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Properties

Label 1-1021-1021.106-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.999 + 0.0280i$
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  + (0.343 − 0.938i)2-s + (0.932 + 0.361i)3-s + (−0.763 − 0.645i)4-s + (0.980 + 0.195i)5-s + (0.659 − 0.751i)6-s + (−0.956 + 0.291i)7-s + (−0.869 + 0.494i)8-s + (0.739 + 0.673i)9-s + (0.521 − 0.853i)10-s + (−0.794 + 0.607i)11-s + (−0.478 − 0.878i)12-s + (0.273 − 0.961i)13-s + (−0.0554 + 0.998i)14-s + (0.843 + 0.536i)15-s + (0.165 + 0.986i)16-s + (−0.297 + 0.954i)17-s + ⋯
L(s)  = 1  + (0.343 − 0.938i)2-s + (0.932 + 0.361i)3-s + (−0.763 − 0.645i)4-s + (0.980 + 0.195i)5-s + (0.659 − 0.751i)6-s + (−0.956 + 0.291i)7-s + (−0.869 + 0.494i)8-s + (0.739 + 0.673i)9-s + (0.521 − 0.853i)10-s + (−0.794 + 0.607i)11-s + (−0.478 − 0.878i)12-s + (0.273 − 0.961i)13-s + (−0.0554 + 0.998i)14-s + (0.843 + 0.536i)15-s + (0.165 + 0.986i)16-s + (−0.297 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.999 + 0.0280i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.999 + 0.0280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.315375268 + 0.03247892539i\)
\(L(\frac12)\) \(\approx\) \(2.315375268 + 0.03247892539i\)
\(L(1)\) \(\approx\) \(1.596540516 - 0.2921158698i\)
\(L(1)\) \(\approx\) \(1.596540516 - 0.2921158698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.343 - 0.938i)T \)
3 \( 1 + (0.932 + 0.361i)T \)
5 \( 1 + (0.980 + 0.195i)T \)
7 \( 1 + (-0.956 + 0.291i)T \)
11 \( 1 + (-0.794 + 0.607i)T \)
13 \( 1 + (0.273 - 0.961i)T \)
17 \( 1 + (-0.297 + 0.954i)T \)
19 \( 1 + (0.952 + 0.303i)T \)
23 \( 1 + (0.771 + 0.636i)T \)
29 \( 1 + (-0.730 + 0.682i)T \)
31 \( 1 + (-0.592 + 0.805i)T \)
37 \( 1 + (0.987 + 0.159i)T \)
41 \( 1 + (-0.952 - 0.303i)T \)
43 \( 1 + (0.875 - 0.483i)T \)
47 \( 1 + (-0.836 - 0.547i)T \)
53 \( 1 + (0.249 + 0.968i)T \)
59 \( 1 + (-0.704 - 0.709i)T \)
61 \( 1 + (0.892 - 0.451i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.869 - 0.494i)T \)
73 \( 1 + (-0.0554 - 0.998i)T \)
79 \( 1 + (0.903 + 0.429i)T \)
83 \( 1 + (-0.779 + 0.626i)T \)
89 \( 1 + (0.881 - 0.473i)T \)
97 \( 1 + (-0.992 - 0.122i)T \)
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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

−21.60559544965946019934236103818, −20.89786662676087387775605862202, −20.23394999101084082835130882984, −18.91835811078583366484375861809, −18.52614946225046690077461601393, −17.733728625328406995109986781834, −16.51905598251629015370915558015, −16.26071511745781974266868528723, −15.257318058192051282784250508565, −14.31900612281053264385157970234, −13.608654396673311867324360645363, −13.29054292898969500996907641011, −12.642338633746854256638740949721, −11.30842951448989760676551401469, −9.74109624031850047741637085355, −9.43894733595242173511432425904, −8.66114160563786018027251497380, −7.62772066347703420200244842465, −6.853471086233347790010253311570, −6.20298345605915054786912804396, −5.22133248203198901512832780179, −4.15869561028336796083938417459, −3.1052500874020747385507037893, −2.43202382379603256657409930504, −0.79648231401849188013217376154, 1.42300651261774581214537369919, 2.32591183407874931029446347092, 3.114787529825843342552519862519, 3.67245228874770186821947181029, 5.11807033757581746475555627792, 5.59301383724347229544277552155, 6.84343244199267504783605216137, 8.05609849713489454085781717292, 9.121500623232051728062697381587, 9.59124848207421665008508069395, 10.37123309360129711512994567450, 10.82881835535644039488921195635, 12.42821117693976038064379003233, 13.062121174002776093820819210678, 13.38576554424633300538346488323, 14.398724530220646180778480732538, 15.146404649994431536008619702780, 15.763324699934364552841294314512, 17.03123215985101437190517272373, 18.15169415714696528289660313495, 18.522000820618671421889228354433, 19.50668292388018258789171728567, 20.218766273095786212276922342018, 20.72109753764922712358338411070, 21.63348575130628185351241244821

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