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Properties

Label 1-1021-1021.154-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.199 - 0.979i$
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  + (−0.128 + 0.991i)2-s + (−0.273 − 0.961i)3-s + (−0.966 − 0.255i)4-s + (−0.366 + 0.930i)5-s + (0.989 − 0.147i)6-s + (0.975 + 0.219i)7-s + (0.378 − 0.925i)8-s + (−0.850 + 0.526i)9-s + (−0.875 − 0.483i)10-s + (−0.225 + 0.974i)11-s + (0.0184 + 0.999i)12-s + (−0.982 − 0.183i)13-s + (−0.343 + 0.938i)14-s + (0.995 + 0.0984i)15-s + (0.869 + 0.494i)16-s + (−0.936 + 0.349i)17-s + ⋯
L(s)  = 1  + (−0.128 + 0.991i)2-s + (−0.273 − 0.961i)3-s + (−0.966 − 0.255i)4-s + (−0.366 + 0.930i)5-s + (0.989 − 0.147i)6-s + (0.975 + 0.219i)7-s + (0.378 − 0.925i)8-s + (−0.850 + 0.526i)9-s + (−0.875 − 0.483i)10-s + (−0.225 + 0.974i)11-s + (0.0184 + 0.999i)12-s + (−0.982 − 0.183i)13-s + (−0.343 + 0.938i)14-s + (0.995 + 0.0984i)15-s + (0.869 + 0.494i)16-s + (−0.936 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.199 - 0.979i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.199 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06920763953 - 0.08467586478i\)
\(L(\frac12)\) \(\approx\) \(0.06920763953 - 0.08467586478i\)
\(L(1)\) \(\approx\) \(0.5866583297 + 0.2170295874i\)
\(L(1)\) \(\approx\) \(0.5866583297 + 0.2170295874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.128 + 0.991i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.366 + 0.930i)T \)
7 \( 1 + (0.975 + 0.219i)T \)
11 \( 1 + (-0.225 + 0.974i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (-0.936 + 0.349i)T \)
19 \( 1 + (0.881 - 0.473i)T \)
23 \( 1 + (-0.862 - 0.505i)T \)
29 \( 1 + (-0.678 - 0.734i)T \)
31 \( 1 + (-0.177 + 0.984i)T \)
37 \( 1 + (-0.927 + 0.372i)T \)
41 \( 1 + (0.881 - 0.473i)T \)
43 \( 1 + (0.116 - 0.993i)T \)
47 \( 1 + (0.985 - 0.171i)T \)
53 \( 1 + (-0.747 + 0.664i)T \)
59 \( 1 + (0.0677 + 0.997i)T \)
61 \( 1 + (-0.641 - 0.767i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (0.378 - 0.925i)T \)
73 \( 1 + (-0.343 - 0.938i)T \)
79 \( 1 + (-0.945 + 0.326i)T \)
83 \( 1 + (-0.696 + 0.717i)T \)
89 \( 1 + (-0.153 - 0.988i)T \)
97 \( 1 + (-0.908 - 0.417i)T \)
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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

−21.66466415953517401687363581534, −21.004476841891279817851798384526, −20.25740309668253564028712175476, −19.93313104224337499409167747935, −18.82159651870656531789944632186, −17.789096324963003347334546054967, −17.2260530049868275614800856365, −16.4324531966498013037541881755, −15.72711584827343511179346102764, −14.52333421998277935802014193010, −13.95602113921316381715954657908, −12.92568361406865237399247329396, −11.91619740864982658674586194824, −11.42079114156151230621403984349, −10.82222812959085174715486212068, −9.71740134487042027761957174048, −9.16877310990165912164432032532, −8.30452206427778545889359992661, −7.60409874755621104679878403546, −5.63670802656327210843104674288, −5.09106986257859140078872451750, −4.29529964004918029379558688298, −3.636145876853704075482199239144, −2.410999365285999932922133537756, −1.18000008863096613758566282467, 0.054394125772515379411149598646, 1.74257018622867573699468714786, 2.62024740644203218216546167203, 4.19164132713562011480813011967, 5.06112443191974066138693649339, 5.906714377934392179357990876411, 6.979322107473979490350072347036, 7.35755013641500405007097092934, 8.0156025838009886683556156377, 8.96673547015427016543783731982, 10.16840459395312680454193480777, 10.96622682394894264949265399043, 12.0083030121758917787301426712, 12.597361249170237711015614967855, 13.87004031336542876900952602216, 14.21069937258726429933490348891, 15.19716583724059893689410654574, 15.59046989717316427599712145227, 16.98589378888474391735008069764, 17.678875840652155867027090315908, 17.99560521876097362247960294698, 18.738837103181148057278568422698, 19.57709724905958038983901437103, 20.33813434957917815517983286986, 21.91389553671248289869191566392

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