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Properties

Label 1-967-967.98-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0808 + 0.996i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.430 − 0.902i)2-s + (−0.668 − 0.744i)3-s + (−0.628 + 0.777i)4-s + (−0.986 − 0.161i)5-s + (−0.383 + 0.923i)6-s + (0.100 + 0.994i)7-s + (0.972 + 0.232i)8-s + (−0.107 + 0.994i)9-s + (0.279 + 0.960i)10-s + (0.993 − 0.116i)11-s + (0.998 − 0.0520i)12-s + (−0.597 + 0.801i)13-s + (0.854 − 0.519i)14-s + (0.538 + 0.842i)15-s + (−0.209 − 0.977i)16-s + (0.763 − 0.646i)17-s + ⋯
L(s)  = 1  + (−0.430 − 0.902i)2-s + (−0.668 − 0.744i)3-s + (−0.628 + 0.777i)4-s + (−0.986 − 0.161i)5-s + (−0.383 + 0.923i)6-s + (0.100 + 0.994i)7-s + (0.972 + 0.232i)8-s + (−0.107 + 0.994i)9-s + (0.279 + 0.960i)10-s + (0.993 − 0.116i)11-s + (0.998 − 0.0520i)12-s + (−0.597 + 0.801i)13-s + (0.854 − 0.519i)14-s + (0.538 + 0.842i)15-s + (−0.209 − 0.977i)16-s + (0.763 − 0.646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0808 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0808 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0808 + 0.996i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0808 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1659935177 + 0.1530764643i\)
\(L(\frac12)\) \(\approx\) \(0.1659935177 + 0.1530764643i\)
\(L(1)\) \(\approx\) \(0.4669357940 - 0.1725875109i\)
\(L(1)\) \(\approx\) \(0.4669357940 - 0.1725875109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.430 - 0.902i)T \)
3 \( 1 + (-0.668 - 0.744i)T \)
5 \( 1 + (-0.986 - 0.161i)T \)
7 \( 1 + (0.100 + 0.994i)T \)
11 \( 1 + (0.993 - 0.116i)T \)
13 \( 1 + (-0.597 + 0.801i)T \)
17 \( 1 + (0.763 - 0.646i)T \)
19 \( 1 + (0.549 + 0.835i)T \)
23 \( 1 + (-0.957 + 0.288i)T \)
29 \( 1 + (-0.844 - 0.536i)T \)
31 \( 1 + (-0.555 - 0.831i)T \)
37 \( 1 + (-0.999 + 0.0325i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (0.266 + 0.963i)T \)
47 \( 1 + (0.994 + 0.103i)T \)
53 \( 1 + (-0.829 + 0.557i)T \)
59 \( 1 + (-0.0812 - 0.996i)T \)
61 \( 1 + (-0.949 - 0.313i)T \)
67 \( 1 + (-0.442 - 0.896i)T \)
71 \( 1 + (0.996 - 0.0779i)T \)
73 \( 1 + (-0.829 - 0.557i)T \)
79 \( 1 + (0.505 + 0.862i)T \)
83 \( 1 + (-0.815 - 0.579i)T \)
89 \( 1 + (0.997 + 0.0649i)T \)
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

−22.09213657786786777532863435137, −20.506089086632073797877249272709, −19.933414648757695910846408042984, −19.24727352802246448018197614757, −18.10629713161817850116934086354, −17.35229929180767493062432553505, −16.84025596995168679905216337971, −16.089098747880438954721670116775, −15.404190478243892032076607463087, −14.63168084460054965139242707761, −14.07284963587540958536334175742, −12.63574893015190922936378060415, −11.83944069536406156253964921859, −10.704514895136483446551390490724, −10.392137125213651177686835531420, −9.35425924900627695808723565138, −8.49001094410790458584581785170, −7.35082088005124783176109310464, −7.02217324067261283206954408923, −5.82726445878617538703194894531, −4.97207509736648257699330090024, −4.06843225471419467511547490765, −3.508352788435766232259783672889, −1.220886626084544548720537600017, −0.1529545118676245044536348051, 1.26734562183163515532417019557, 2.08674992780861193657569703064, 3.27716811821530529148615458339, 4.27948045689479591801376829265, 5.22852049618630314783686400013, 6.29899641637633173581936892693, 7.5809105281801705931360947981, 7.89092522035400404921468183453, 9.08580764450542714963594053197, 9.70494318624892481983692122623, 11.08103432618447751697845278739, 11.70185180040590662641907062710, 12.07647669369396616418917399688, 12.60091791798974035862848272115, 13.85757562178733189139398018270, 14.58187243085268546361747172067, 15.965353888837047183142253904069, 16.62858280961979330706230505647, 17.27919151437588989775859666246, 18.47024127488792502967375879005, 18.693664754803953867906705654422, 19.42748972520074512139116753782, 20.12253359436826679099465169287, 21.12122117560905087264494379669, 22.11967031587313870092694147555

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