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Properties

Label 1-967-967.81-r0-0-0
Degree $1$
Conductor $967$
Sign $0.231 - 0.972i$
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  + (0.981 + 0.193i)2-s + (−0.864 − 0.502i)3-s + (0.924 + 0.380i)4-s + (0.993 − 0.116i)5-s + (−0.750 − 0.660i)6-s + (0.00975 − 0.999i)7-s + (0.833 + 0.552i)8-s + (0.494 + 0.869i)9-s + (0.996 + 0.0779i)10-s + (−0.957 + 0.288i)11-s + (−0.608 − 0.793i)12-s + (−0.957 − 0.288i)13-s + (0.203 − 0.979i)14-s + (−0.917 − 0.398i)15-s + (0.710 + 0.703i)16-s + (−0.184 − 0.982i)17-s + ⋯
L(s)  = 1  + (0.981 + 0.193i)2-s + (−0.864 − 0.502i)3-s + (0.924 + 0.380i)4-s + (0.993 − 0.116i)5-s + (−0.750 − 0.660i)6-s + (0.00975 − 0.999i)7-s + (0.833 + 0.552i)8-s + (0.494 + 0.869i)9-s + (0.996 + 0.0779i)10-s + (−0.957 + 0.288i)11-s + (−0.608 − 0.793i)12-s + (−0.957 − 0.288i)13-s + (0.203 − 0.979i)14-s + (−0.917 − 0.398i)15-s + (0.710 + 0.703i)16-s + (−0.184 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.231 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.722785618 - 1.360314524i\)
\(L(\frac12)\) \(\approx\) \(1.722785618 - 1.360314524i\)
\(L(1)\) \(\approx\) \(1.547439245 - 0.4054299652i\)
\(L(1)\) \(\approx\) \(1.547439245 - 0.4054299652i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.981 + 0.193i)T \)
3 \( 1 + (-0.864 - 0.502i)T \)
5 \( 1 + (0.993 - 0.116i)T \)
7 \( 1 + (0.00975 - 0.999i)T \)
11 \( 1 + (-0.957 + 0.288i)T \)
13 \( 1 + (-0.957 - 0.288i)T \)
17 \( 1 + (-0.184 - 0.982i)T \)
19 \( 1 + (-0.145 - 0.989i)T \)
23 \( 1 + (-0.668 - 0.744i)T \)
29 \( 1 + (0.987 - 0.155i)T \)
31 \( 1 + (0.353 - 0.935i)T \)
37 \( 1 + (-0.822 + 0.568i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (0.592 - 0.805i)T \)
47 \( 1 + (-0.260 - 0.965i)T \)
53 \( 1 + (-0.576 + 0.816i)T \)
59 \( 1 + (-0.998 - 0.0585i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (0.353 + 0.935i)T \)
71 \( 1 + (0.981 - 0.193i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (0.874 + 0.485i)T \)
83 \( 1 + (-0.477 + 0.878i)T \)
89 \( 1 + (0.353 + 0.935i)T \)
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

−21.82628059536396902099305456405, −21.24483583865455653983528353105, −21.06486382462947659086513003013, −19.62928831418522884138304176486, −18.79114231710700524835300483492, −17.82960042874650951850818185340, −17.19994424603623475744656416187, −16.0579453087730085778968323526, −15.712504965090285721872551170701, −14.62238823004470497295152641212, −14.138859612605423763195325386318, −12.75557617838703599355480506927, −12.54372826874161922094018905804, −11.59601324536166481260617570689, −10.6143010880469535260743639455, −10.13790544177956729047728615998, −9.25825763759150000266968026624, −7.88386198938603839619438932066, −6.55217884728630798896686943956, −5.97968906372443206995739944421, −5.330833057809567198246725881033, −4.68647216882845956045151270184, −3.43184725040778005124967110002, −2.42795593442908304797955288445, −1.5634028507778852887628558396, 0.71290556714879069066049252630, 2.15272905868211087498067936682, 2.730041368876044320700461289465, 4.50105081741198672164136451719, 4.89726399617280115558332067805, 5.75479483544919740930786014917, 6.72409752072990869563507915860, 7.21756216148366928395818792911, 8.11937805490802128942485465820, 9.83736156124057692833710095792, 10.44518087443013234279896911082, 11.19439255254718427317317166891, 12.260240575057735519066395155540, 12.842744569771326040572119545918, 13.68717507216554283856458335572, 13.96030887585660543793801472569, 15.26126234853481622694964138846, 16.12716998508670082209672714698, 16.88743849790051536047677889369, 17.50084454260571179689375715111, 18.09779033214317537126489246673, 19.3508356790962297193434615922, 20.31759380042183096890460043404, 20.88228893095645170575167025121, 21.8679897499915592227669460561

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