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Properties

Label 1-967-967.776-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: $\chi_{967} (776, \cdot )$
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

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

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