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Properties

Label 1-967-967.113-r0-0-0
Degree $1$
Conductor $967$
Sign $0.915 + 0.402i$
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  + (0.934 + 0.356i)2-s + (0.962 − 0.269i)3-s + (0.746 + 0.665i)4-s + (0.803 − 0.595i)5-s + (0.995 + 0.0909i)6-s + (0.291 + 0.956i)7-s + (0.460 + 0.887i)8-s + (0.854 − 0.519i)9-s + (0.962 − 0.269i)10-s + (0.854 − 0.519i)11-s + (0.898 + 0.439i)12-s + (−0.877 + 0.480i)13-s + (−0.0682 + 0.997i)14-s + (0.613 − 0.789i)15-s + (0.113 + 0.993i)16-s + (−0.990 + 0.136i)17-s + ⋯
L(s)  = 1  + (0.934 + 0.356i)2-s + (0.962 − 0.269i)3-s + (0.746 + 0.665i)4-s + (0.803 − 0.595i)5-s + (0.995 + 0.0909i)6-s + (0.291 + 0.956i)7-s + (0.460 + 0.887i)8-s + (0.854 − 0.519i)9-s + (0.962 − 0.269i)10-s + (0.854 − 0.519i)11-s + (0.898 + 0.439i)12-s + (−0.877 + 0.480i)13-s + (−0.0682 + 0.997i)14-s + (0.613 − 0.789i)15-s + (0.113 + 0.993i)16-s + (−0.990 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.418393133 + 0.9276320672i\)
\(L(\frac12)\) \(\approx\) \(4.418393133 + 0.9276320672i\)
\(L(1)\) \(\approx\) \(2.788761677 + 0.4209347724i\)
\(L(1)\) \(\approx\) \(2.788761677 + 0.4209347724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.934 + 0.356i)T \)
3 \( 1 + (0.962 - 0.269i)T \)
5 \( 1 + (0.803 - 0.595i)T \)
7 \( 1 + (0.291 + 0.956i)T \)
11 \( 1 + (0.854 - 0.519i)T \)
13 \( 1 + (-0.877 + 0.480i)T \)
17 \( 1 + (-0.990 + 0.136i)T \)
19 \( 1 + (-0.715 - 0.699i)T \)
23 \( 1 + (0.203 - 0.979i)T \)
29 \( 1 + (0.854 + 0.519i)T \)
31 \( 1 + (-0.998 - 0.0455i)T \)
37 \( 1 + (-0.877 + 0.480i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.377 + 0.926i)T \)
47 \( 1 + (0.613 - 0.789i)T \)
53 \( 1 + (-0.949 + 0.313i)T \)
59 \( 1 + (-0.949 - 0.313i)T \)
61 \( 1 + (-0.419 - 0.907i)T \)
67 \( 1 + (0.460 - 0.887i)T \)
71 \( 1 + (-0.775 - 0.631i)T \)
73 \( 1 + (-0.949 - 0.313i)T \)
79 \( 1 + (-0.949 - 0.313i)T \)
83 \( 1 + (0.995 + 0.0909i)T \)
89 \( 1 + (0.538 + 0.842i)T \)
97 \( 1 + T \)
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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

−21.76412757379775598235787470575, −20.85347129832329976390590812662, −20.3153584009879517637149011273, −19.57670124081641155834965554881, −18.98656301586187216933020526710, −17.6385750501159802063126418316, −17.055374534291296393967197800172, −15.781376472909979291079202783503, −14.99440244104858873830177722375, −14.37993649796477711093108079027, −13.89465407462660567586026774445, −13.16203591075627325845973553971, −12.32182141718231202134492832348, −11.099299087540442032056482115204, −10.34872932478329370265082596390, −9.85128645373258782900740210686, −8.88037919542996731653759252268, −7.30584488736841113838209604614, −7.101982681741240313316086677319, −5.87977308504812392686094884321, −4.738277627409323768142825149491, −4.03249314949358552273693283748, −3.184469072964223385092634195217, −2.15043068608434095247539398191, −1.559784027279162512794213694200, 1.66576445303412998145226203885, 2.28279404150595115830241518246, 3.13520147084941713889026866623, 4.481988514457939441681940731967, 4.93279393785048104827038214841, 6.34988637898370610292306443637, 6.620240341111292741976767685691, 7.96022268194910749070802296185, 8.96046834009267488977342461786, 9.04736636479060228925659443672, 10.58383051685917922386289950409, 11.762748281234020713672719297712, 12.49725199997352039006603522539, 13.080769590005649258258356387218, 13.97533049515818686190511466381, 14.520153482584268949653607537876, 15.18200298913645864579411004978, 16.08949787923045044318849960089, 16.99900577468062877408151834353, 17.714927964770243507966238554469, 18.76868866232525127766126267415, 19.75324723277122410357780121428, 20.27327186242013244931353418865, 21.29534634087840212845742419368, 21.76341273180541745626679446721

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