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Properties

Label 1-1021-1021.114-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.999 + 0.00832i$
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  + (−0.412 + 0.911i)2-s + (0.445 + 0.895i)3-s + (−0.659 − 0.751i)4-s + (0.467 − 0.883i)5-s + (−0.999 + 0.0369i)6-s + (−0.0554 + 0.998i)7-s + (0.956 − 0.291i)8-s + (−0.602 + 0.798i)9-s + (0.612 + 0.790i)10-s + (−0.434 + 0.900i)11-s + (0.378 − 0.925i)12-s + (0.739 − 0.673i)13-s + (−0.886 − 0.462i)14-s + (0.999 + 0.0246i)15-s + (−0.128 + 0.991i)16-s + (−0.641 + 0.767i)17-s + ⋯
L(s)  = 1  + (−0.412 + 0.911i)2-s + (0.445 + 0.895i)3-s + (−0.659 − 0.751i)4-s + (0.467 − 0.883i)5-s + (−0.999 + 0.0369i)6-s + (−0.0554 + 0.998i)7-s + (0.956 − 0.291i)8-s + (−0.602 + 0.798i)9-s + (0.612 + 0.790i)10-s + (−0.434 + 0.900i)11-s + (0.378 − 0.925i)12-s + (0.739 − 0.673i)13-s + (−0.886 − 0.462i)14-s + (0.999 + 0.0246i)15-s + (−0.128 + 0.991i)16-s + (−0.641 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.999 + 0.00832i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.999 + 0.00832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004694341596 + 1.127240800i\)
\(L(\frac12)\) \(\approx\) \(0.004694341596 + 1.127240800i\)
\(L(1)\) \(\approx\) \(0.6530170195 + 0.6798312292i\)
\(L(1)\) \(\approx\) \(0.6530170195 + 0.6798312292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.412 + 0.911i)T \)
3 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (0.467 - 0.883i)T \)
7 \( 1 + (-0.0554 + 0.998i)T \)
11 \( 1 + (-0.434 + 0.900i)T \)
13 \( 1 + (0.739 - 0.673i)T \)
17 \( 1 + (-0.641 + 0.767i)T \)
19 \( 1 + (0.992 - 0.122i)T \)
23 \( 1 + (-0.794 + 0.607i)T \)
29 \( 1 + (-0.836 + 0.547i)T \)
31 \( 1 + (0.423 - 0.905i)T \)
37 \( 1 + (0.771 + 0.636i)T \)
41 \( 1 + (0.992 - 0.122i)T \)
43 \( 1 + (0.355 + 0.934i)T \)
47 \( 1 + (0.0431 + 0.999i)T \)
53 \( 1 + (-0.823 - 0.567i)T \)
59 \( 1 + (-0.366 + 0.930i)T \)
61 \( 1 + (0.843 - 0.536i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.956 - 0.291i)T \)
73 \( 1 + (-0.886 + 0.462i)T \)
79 \( 1 + (-0.763 - 0.645i)T \)
83 \( 1 + (0.552 - 0.833i)T \)
89 \( 1 + (-0.908 + 0.417i)T \)
97 \( 1 + (-0.779 + 0.626i)T \)
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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

−20.93983973206406166529401969961, −20.39224510049468513528873117738, −19.575831442433150698367769712982, −18.82215889701732782317024666951, −18.25838724163888857353505276342, −17.76657577990960032594349329191, −16.76658862266963167005385932717, −15.89362918862664974097437387901, −14.31249804641095170507099628261, −13.76423177529983685220466427688, −13.55311729938486260414685602406, −12.47957128950422332104707148383, −11.29860302979877048912871529024, −11.06700146400636963249098706408, −9.9595773741432037949933329461, −9.20807013283816075228037682878, −8.24521667801642862812987582384, −7.440705209336512018282409498584, −6.74580311494167441839311725929, −5.70774275033532283575287322922, −4.12259804040997002922812246246, −3.31524876334313592496462710859, −2.5491895854955603321324587770, −1.617047113980177874609377816505, −0.527126258480104448487322793370, 1.45920342036598005279245471972, 2.53330981640438129056439826066, 3.9849829257736736878025834460, 4.83132968390199995922695590631, 5.59095765400174855336404085930, 6.12675592611958771457542804013, 7.788778331337538493134189054427, 8.21073681899717465004339995544, 9.2503274101722903254598955836, 9.503452039152972748826957157662, 10.39296635962376503159284508936, 11.4687868901448509378635957097, 12.8735035174291303737704393327, 13.30432724740596267851903729113, 14.421095795525528778521917926458, 15.13473615942664791267235763162, 15.890890174161254889853016730409, 16.11020645807192739742547181980, 17.37427452686357914725434545411, 17.79186445127404046269873999014, 18.706297670059069277307395776126, 19.8163335305769616583267505745, 20.352473661240800334461999103664, 21.177032382438764295928879944824, 22.13331044966123705471252625126

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