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Properties

Label 1-967-967.202-r0-0-0
Degree $1$
Conductor $967$
Sign $0.886 - 0.463i$
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  + (−0.247 − 0.968i)2-s + (0.203 + 0.979i)3-s + (−0.877 + 0.480i)4-s + (−0.998 − 0.0455i)5-s + (0.898 − 0.439i)6-s + (0.995 − 0.0909i)7-s + (0.682 + 0.730i)8-s + (−0.917 + 0.398i)9-s + (0.203 + 0.979i)10-s + (−0.917 + 0.398i)11-s + (−0.648 − 0.761i)12-s + (0.803 − 0.595i)13-s + (−0.334 − 0.942i)14-s + (−0.158 − 0.987i)15-s + (0.538 − 0.842i)16-s + (−0.775 − 0.631i)17-s + ⋯
L(s)  = 1  + (−0.247 − 0.968i)2-s + (0.203 + 0.979i)3-s + (−0.877 + 0.480i)4-s + (−0.998 − 0.0455i)5-s + (0.898 − 0.439i)6-s + (0.995 − 0.0909i)7-s + (0.682 + 0.730i)8-s + (−0.917 + 0.398i)9-s + (0.203 + 0.979i)10-s + (−0.917 + 0.398i)11-s + (−0.648 − 0.761i)12-s + (0.803 − 0.595i)13-s + (−0.334 − 0.942i)14-s + (−0.158 − 0.987i)15-s + (0.538 − 0.842i)16-s + (−0.775 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.886 - 0.463i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.886 - 0.463i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9911103305 - 0.2434583708i\)
\(L(\frac12)\) \(\approx\) \(0.9911103305 - 0.2434583708i\)
\(L(1)\) \(\approx\) \(0.8268885408 - 0.1332929672i\)
\(L(1)\) \(\approx\) \(0.8268885408 - 0.1332929672i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.247 - 0.968i)T \)
3 \( 1 + (0.203 + 0.979i)T \)
5 \( 1 + (-0.998 - 0.0455i)T \)
7 \( 1 + (0.995 - 0.0909i)T \)
11 \( 1 + (-0.917 + 0.398i)T \)
13 \( 1 + (0.803 - 0.595i)T \)
17 \( 1 + (-0.775 - 0.631i)T \)
19 \( 1 + (0.746 - 0.665i)T \)
23 \( 1 + (0.854 + 0.519i)T \)
29 \( 1 + (-0.917 - 0.398i)T \)
31 \( 1 + (-0.974 + 0.225i)T \)
37 \( 1 + (0.803 - 0.595i)T \)
41 \( 1 + (-0.0682 + 0.997i)T \)
43 \( 1 + (0.934 + 0.356i)T \)
47 \( 1 + (-0.158 - 0.987i)T \)
53 \( 1 + (0.0227 - 0.999i)T \)
59 \( 1 + (0.0227 + 0.999i)T \)
61 \( 1 + (-0.829 - 0.557i)T \)
67 \( 1 + (0.682 - 0.730i)T \)
71 \( 1 + (0.962 - 0.269i)T \)
73 \( 1 + (0.0227 + 0.999i)T \)
79 \( 1 + (0.0227 + 0.999i)T \)
83 \( 1 + (0.898 - 0.439i)T \)
89 \( 1 + (0.291 + 0.956i)T \)
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

−22.096341480842404426724039413647, −20.78916732659249326833691643165, −20.14892667033900580316924325178, −18.94932158345147915964342924373, −18.66935346675321304297281747668, −18.04359364458437949221005640460, −17.09092159023944841834588268422, −16.28727944770365814542795660384, −15.42275571859426981068988878575, −14.71701561045735189215731919561, −14.014799381069995726881879299, −13.16559207143891699119706047926, −12.408752932655431739883098866274, −11.19500659916678983558363114812, −10.8230610127384725213123801952, −9.03825920301805512563129321883, −8.56109404790368791569411307605, −7.7329727523852132949922778296, −7.36141829896579541279443773103, −6.26282326454587229665991778826, −5.434173211140261684829022908252, −4.39778275117949633023456146776, −3.38567367206054901109395906291, −1.89715225337818586228367288002, −0.82346082298586891235212298453, 0.708911079861569177187519279890, 2.23606827488619988525153062668, 3.171614140991197648143125398030, 3.95525654795243508827943061621, 4.85754072605022543222910803079, 5.30611554747758828227927019141, 7.43516549523569215640907379704, 7.98124385314837123565819569725, 8.83798844369484739806712791365, 9.52814418470795426940589161062, 10.7196801453425802951199428571, 11.13861160756776883667583369022, 11.58990969987037809833114312547, 12.88219228337559040154543570622, 13.570836137824943537018265649652, 14.660366062808910427157471345638, 15.36560066409341065871032999610, 16.06535390154674592832344736302, 17.03727192262343170083259820385, 18.04609666962924436981703855504, 18.43538045646857949466155088410, 19.86625312658430540827953167451, 20.0462130836822760206120748493, 20.87500253164473152326211622402, 21.343454523603346312986465504793

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