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Properties

Label 1-967-967.212-r0-0-0
Degree $1$
Conductor $967$
Sign $0.955 + 0.293i$
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  + (0.592 + 0.805i)2-s + (−0.799 − 0.600i)3-s + (−0.297 + 0.954i)4-s + (−0.371 + 0.928i)5-s + (0.00975 − 0.999i)6-s + (0.353 − 0.935i)7-s + (−0.945 + 0.325i)8-s + (0.279 + 0.960i)9-s + (−0.967 + 0.250i)10-s + (0.165 − 0.986i)11-s + (0.811 − 0.584i)12-s + (0.165 + 0.986i)13-s + (0.962 − 0.269i)14-s + (0.854 − 0.519i)15-s + (−0.822 − 0.568i)16-s + (−0.544 − 0.838i)17-s + ⋯
L(s)  = 1  + (0.592 + 0.805i)2-s + (−0.799 − 0.600i)3-s + (−0.297 + 0.954i)4-s + (−0.371 + 0.928i)5-s + (0.00975 − 0.999i)6-s + (0.353 − 0.935i)7-s + (−0.945 + 0.325i)8-s + (0.279 + 0.960i)9-s + (−0.967 + 0.250i)10-s + (0.165 − 0.986i)11-s + (0.811 − 0.584i)12-s + (0.165 + 0.986i)13-s + (0.962 − 0.269i)14-s + (0.854 − 0.519i)15-s + (−0.822 − 0.568i)16-s + (−0.544 − 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.955 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.242788527 + 0.1865327868i\)
\(L(\frac12)\) \(\approx\) \(1.242788527 + 0.1865327868i\)
\(L(1)\) \(\approx\) \(0.9836849458 + 0.2855938428i\)
\(L(1)\) \(\approx\) \(0.9836849458 + 0.2855938428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.592 + 0.805i)T \)
3 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (-0.371 + 0.928i)T \)
7 \( 1 + (0.353 - 0.935i)T \)
11 \( 1 + (0.165 - 0.986i)T \)
13 \( 1 + (0.165 + 0.986i)T \)
17 \( 1 + (-0.544 - 0.838i)T \)
19 \( 1 + (0.763 - 0.646i)T \)
23 \( 1 + (-0.932 + 0.362i)T \)
29 \( 1 + (0.874 + 0.485i)T \)
31 \( 1 + (0.710 - 0.703i)T \)
37 \( 1 + (0.924 - 0.380i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (-0.995 + 0.0974i)T \)
47 \( 1 + (0.316 + 0.948i)T \)
53 \( 1 + (0.682 - 0.730i)T \)
59 \( 1 + (0.560 - 0.828i)T \)
61 \( 1 + (-0.775 - 0.631i)T \)
67 \( 1 + (0.710 + 0.703i)T \)
71 \( 1 + (0.592 - 0.805i)T \)
73 \( 1 + (0.682 + 0.730i)T \)
79 \( 1 + (0.996 - 0.0779i)T \)
83 \( 1 + (0.425 + 0.905i)T \)
89 \( 1 + (0.710 + 0.703i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−21.6389951122626464684188739425, −21.07629442107104050884719306469, −20.15503177659532737124126202093, −19.86720206815468409654477448144, −18.40290909173278203332430860443, −17.94530537012075474851176362818, −17.05144254766961006732473602227, −15.91517411830717560866616908508, −15.32237457421617584879137387889, −14.82765672589024925090054886297, −13.47544311076780540952942648836, −12.53133090384879863432141802766, −12.066681701188104181489337363381, −11.60690219658141020394756160990, −10.37888781738626047462559511594, −9.9030878381325155262377244650, −8.89307474791592838031401573008, −8.07866830622618650909372811272, −6.40938286104586041900390019887, −5.65440604780322579414730025820, −4.8980888782645958612378697547, −4.328363488196061042512922860286, −3.33736964246478793631881285891, −1.99653740246192817427651654988, −0.97218344788895581619042432287, 0.63410063513984668524873800132, 2.30993296238720902829187879965, 3.49204780727474181698419061675, 4.36972128790787659213222320921, 5.24887437729922654550242317846, 6.46323495089301530689632235770, 6.720339703373158861842320129347, 7.59517624381933903934871082230, 8.26444949129987629411781115416, 9.64099992333717770894049902073, 10.93600333617685948394095318294, 11.472890128743195041656769114, 12.0047960662940925471948621571, 13.465300814188275810984577578297, 13.768244639308010727567856035663, 14.37187704194579114721245345152, 15.68536686035100566535036930723, 16.20606953795165199397990055031, 16.964149597013738455689578434552, 17.84606985245862042773165927913, 18.32543875584212396200757970629, 19.255624715594527825260434563, 20.19733038703221762922845673591, 21.48097548437396660317222084702, 22.02156373876498617744709821315

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