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Properties

Label 1-967-967.203-r0-0-0
Degree $1$
Conductor $967$
Sign $0.631 - 0.775i$
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  + (−0.911 − 0.410i)2-s + (0.909 − 0.416i)3-s + (0.663 + 0.748i)4-s + (0.266 − 0.963i)5-s + (−0.999 + 0.00650i)6-s + (0.692 + 0.721i)7-s + (−0.297 − 0.954i)8-s + (0.653 − 0.756i)9-s + (−0.638 + 0.769i)10-s + (0.592 + 0.805i)11-s + (0.914 + 0.404i)12-s + (0.401 + 0.915i)13-s + (−0.334 − 0.942i)14-s + (−0.158 − 0.987i)15-s + (−0.120 + 0.992i)16-s + (0.787 − 0.615i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.410i)2-s + (0.909 − 0.416i)3-s + (0.663 + 0.748i)4-s + (0.266 − 0.963i)5-s + (−0.999 + 0.00650i)6-s + (0.692 + 0.721i)7-s + (−0.297 − 0.954i)8-s + (0.653 − 0.756i)9-s + (−0.638 + 0.769i)10-s + (0.592 + 0.805i)11-s + (0.914 + 0.404i)12-s + (0.401 + 0.915i)13-s + (−0.334 − 0.942i)14-s + (−0.158 − 0.987i)15-s + (−0.120 + 0.992i)16-s + (0.787 − 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.570424970 - 0.7469226395i\)
\(L(\frac12)\) \(\approx\) \(1.570424970 - 0.7469226395i\)
\(L(1)\) \(\approx\) \(1.133873163 - 0.3803342852i\)
\(L(1)\) \(\approx\) \(1.133873163 - 0.3803342852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.911 - 0.410i)T \)
3 \( 1 + (0.909 - 0.416i)T \)
5 \( 1 + (0.266 - 0.963i)T \)
7 \( 1 + (0.692 + 0.721i)T \)
11 \( 1 + (0.592 + 0.805i)T \)
13 \( 1 + (0.401 + 0.915i)T \)
17 \( 1 + (0.787 - 0.615i)T \)
19 \( 1 + (-0.0552 - 0.998i)T \)
23 \( 1 + (-0.696 + 0.717i)T \)
29 \( 1 + (-0.184 + 0.982i)T \)
31 \( 1 + (-0.00325 - 0.999i)T \)
37 \( 1 + (0.966 + 0.257i)T \)
41 \( 1 + (-0.0682 + 0.997i)T \)
43 \( 1 + (-0.555 + 0.831i)T \)
47 \( 1 + (0.672 - 0.739i)T \)
53 \( 1 + (0.0227 - 0.999i)T \)
59 \( 1 + (0.795 + 0.605i)T \)
61 \( 1 + (-0.829 - 0.557i)T \)
67 \( 1 + (-0.864 - 0.502i)T \)
71 \( 1 + (0.811 + 0.584i)T \)
73 \( 1 + (0.0227 + 0.999i)T \)
79 \( 1 + (-0.454 - 0.890i)T \)
83 \( 1 + (0.228 + 0.973i)T \)
89 \( 1 + (0.867 - 0.497i)T \)
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

−21.636547094573761324004497499538, −20.8482061576522072237704254752, −20.23849796816005066014215632169, −19.344106860886889372275300795471, −18.768909433485267269616064543067, −18.02157477608060911707387780729, −17.11280822482627746350684922544, −16.41550392633355237723489828514, −15.46776072064843040765767999539, −14.67361084650282539455435109630, −14.24190054891120487653215919812, −13.586399177889349828323555119666, −12.030744169357220739390906119757, −10.74692933471767246466269835647, −10.56161735507822025998653083576, −9.793397756380660020045554173513, −8.685859515752452556567890301662, −7.97943812736078036113077588061, −7.48036170870436909485540581610, −6.307891656322754226920905157967, −5.564580881434899866591000697923, −4.03853611797981523290286372410, −3.24622141665545321575611892412, −2.13449532388348241668196432620, −1.16453556355750249287114390650, 1.17564004784363656029040434797, 1.77418304845881189028700065556, 2.586169428861333833084264473669, 3.851897981940287040785361471332, 4.78514447643846428625628639592, 6.17840398472870075471070660383, 7.19867458343442270783480794886, 7.995647703720643544802803089120, 8.73372102907524467821849664334, 9.39500731921319770030387240156, 9.78619432922169266178470331440, 11.47636997786386032952156222379, 11.86736920232161047246493874412, 12.73850446418189500534074163082, 13.509426457691941881262197910200, 14.538683775508982385607401190699, 15.3581325076925889693933020956, 16.2541115290500171219489298455, 17.06179562461150928455711716520, 18.04037980000934049172198652291, 18.35131277833422147121480399502, 19.412140078840368368296373387442, 20.05633749732512200875510537796, 20.61435166317444334404310001645, 21.38017974360782497999467355236

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