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Properties

Label 1-967-967.439-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.445 + 0.895i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.0357 − 0.999i)2-s + (−0.967 − 0.250i)3-s + (−0.997 − 0.0714i)4-s + (0.228 + 0.973i)5-s + (−0.285 + 0.958i)6-s + (−0.235 + 0.971i)7-s + (−0.107 + 0.994i)8-s + (0.874 + 0.485i)9-s + (0.981 − 0.193i)10-s + (−0.668 + 0.744i)11-s + (0.947 + 0.319i)12-s + (−0.310 + 0.950i)13-s + (0.962 + 0.269i)14-s + (0.0227 − 0.999i)15-s + (0.989 + 0.142i)16-s + (0.316 + 0.948i)17-s + ⋯
L(s)  = 1  + (0.0357 − 0.999i)2-s + (−0.967 − 0.250i)3-s + (−0.997 − 0.0714i)4-s + (0.228 + 0.973i)5-s + (−0.285 + 0.958i)6-s + (−0.235 + 0.971i)7-s + (−0.107 + 0.994i)8-s + (0.874 + 0.485i)9-s + (0.981 − 0.193i)10-s + (−0.668 + 0.744i)11-s + (0.947 + 0.319i)12-s + (−0.310 + 0.950i)13-s + (0.962 + 0.269i)14-s + (0.0227 − 0.999i)15-s + (0.989 + 0.142i)16-s + (0.316 + 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.445 + 0.895i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3052004108 + 0.4929331751i\)
\(L(\frac12)\) \(\approx\) \(0.3052004108 + 0.4929331751i\)
\(L(1)\) \(\approx\) \(0.6517656761 + 0.02261800463i\)
\(L(1)\) \(\approx\) \(0.6517656761 + 0.02261800463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.0357 - 0.999i)T \)
3 \( 1 + (-0.967 - 0.250i)T \)
5 \( 1 + (0.228 + 0.973i)T \)
7 \( 1 + (-0.235 + 0.971i)T \)
11 \( 1 + (-0.668 + 0.744i)T \)
13 \( 1 + (-0.310 + 0.950i)T \)
17 \( 1 + (0.316 + 0.948i)T \)
19 \( 1 + (0.994 + 0.103i)T \)
23 \( 1 + (-0.864 + 0.502i)T \)
29 \( 1 + (0.924 + 0.380i)T \)
31 \( 1 + (-0.597 - 0.801i)T \)
37 \( 1 + (0.549 + 0.835i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (-0.272 - 0.962i)T \)
47 \( 1 + (0.795 - 0.605i)T \)
53 \( 1 + (-0.974 + 0.225i)T \)
59 \( 1 + (-0.783 + 0.620i)T \)
61 \( 1 + (0.934 + 0.356i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (-0.883 - 0.468i)T \)
73 \( 1 + (-0.974 - 0.225i)T \)
79 \( 1 + (0.975 - 0.219i)T \)
83 \( 1 + (0.997 + 0.0649i)T \)
89 \( 1 + (-0.395 - 0.918i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.78794315142214569875857149721, −20.77552342735938336478731800870, −20.08863758253096014549744419810, −18.85192863077449112687938178700, −17.79838750685033752611415829475, −17.58847351850356617158375273691, −16.47156639910887903567674763190, −16.18689690400044356774228712143, −15.65822070360127160420371837676, −14.23891534950977689624368022103, −13.57907454613656810958144677458, −12.792631339208140420994371996509, −12.12989883936386915232386626167, −10.85368110188411898821504608187, −9.985655889161926009022349974227, −9.44607020021138673336239842268, −8.17365129055258088686442521935, −7.54347991822725410837873938248, −6.5439095307685812881082699945, −5.58139066738660059890531358744, −5.119108134778519084696245402647, −4.29295809593245398746387993196, −3.22543288977597216313335003046, −1.00863243030913040501876663207, −0.35050076869871125281614988340, 1.638721892805518706211487995925, 2.24741091208361567401290288854, 3.32567607591453651429922148314, 4.47087513515869210280050284240, 5.44353315972831812137211623127, 6.10137396312398932623647177614, 7.16826535512504390092486372153, 8.14051267622476417821692373618, 9.532369992672151249129868438407, 10.012453768443585279095784791114, 10.77359006430234818753176898260, 11.83799058261336101781511903710, 11.99036914029406268082018583634, 13.02337531330977357395189361483, 13.81141686894516668967834497578, 14.81638228454549006851811962226, 15.58495563167382165530307738503, 16.72338119710661999059071954586, 17.670592256160651244183165318834, 18.25517998116572229156108216521, 18.732436112609338165687697125334, 19.40224151924090130670538511258, 20.521474123923888235705755247486, 21.65977838940605621182512760023, 21.8829721488725463591233652030

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