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Properties

Label 1-967-967.213-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.420 + 0.907i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.260 + 0.965i)2-s + (−0.763 + 0.646i)3-s + (−0.864 − 0.502i)4-s + (−0.987 − 0.155i)5-s + (−0.425 − 0.905i)6-s + (0.511 + 0.859i)7-s + (0.710 − 0.703i)8-s + (0.165 − 0.986i)9-s + (0.407 − 0.913i)10-s + (0.924 + 0.380i)11-s + (0.984 − 0.174i)12-s + (−0.924 + 0.380i)13-s + (−0.962 + 0.269i)14-s + (0.854 − 0.519i)15-s + (0.494 + 0.869i)16-s + (−0.696 + 0.717i)17-s + ⋯
L(s)  = 1  + (−0.260 + 0.965i)2-s + (−0.763 + 0.646i)3-s + (−0.864 − 0.502i)4-s + (−0.987 − 0.155i)5-s + (−0.425 − 0.905i)6-s + (0.511 + 0.859i)7-s + (0.710 − 0.703i)8-s + (0.165 − 0.986i)9-s + (0.407 − 0.913i)10-s + (0.924 + 0.380i)11-s + (0.984 − 0.174i)12-s + (−0.924 + 0.380i)13-s + (−0.962 + 0.269i)14-s + (0.854 − 0.519i)15-s + (0.494 + 0.869i)16-s + (−0.696 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.420 + 0.907i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.420 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3448333607 + 0.5398821112i\)
\(L(\frac12)\) \(\approx\) \(0.3448333607 + 0.5398821112i\)
\(L(1)\) \(\approx\) \(0.4256177506 + 0.3428428940i\)
\(L(1)\) \(\approx\) \(0.4256177506 + 0.3428428940i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.260 + 0.965i)T \)
3 \( 1 + (-0.763 + 0.646i)T \)
5 \( 1 + (-0.987 - 0.155i)T \)
7 \( 1 + (0.511 + 0.859i)T \)
11 \( 1 + (0.924 + 0.380i)T \)
13 \( 1 + (-0.924 + 0.380i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (-0.981 - 0.193i)T \)
23 \( 1 + (-0.560 - 0.828i)T \)
29 \( 1 + (0.668 - 0.744i)T \)
31 \( 1 + (-0.844 - 0.536i)T \)
37 \( 1 + (-0.279 + 0.960i)T \)
41 \( 1 + (0.775 + 0.631i)T \)
43 \( 1 + (-0.316 - 0.948i)T \)
47 \( 1 + (-0.938 - 0.344i)T \)
53 \( 1 + (0.682 - 0.730i)T \)
59 \( 1 + (0.996 - 0.0779i)T \)
61 \( 1 + (-0.775 - 0.631i)T \)
67 \( 1 + (0.844 - 0.536i)T \)
71 \( 1 + (-0.260 - 0.965i)T \)
73 \( 1 + (0.682 + 0.730i)T \)
79 \( 1 + (0.932 + 0.362i)T \)
83 \( 1 + (0.787 - 0.615i)T \)
89 \( 1 + (0.844 - 0.536i)T \)
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

−21.39191629362919629513265116124, −20.15341121091567746016092116219, −19.59873108921124839730049910783, −19.277067813704338098720582680082, −18.02201827794842041407419495026, −17.68045778557990435505028107799, −16.7686963088571660770560513766, −16.149294095183622218986026596322, −14.648796364073828576654540569552, −14.00982169206893808147848893580, −13.00715568858357727994462845580, −12.24986510648896009298169310035, −11.61893852941410998528870058291, −10.959614454688348105874025444095, −10.41052758936213787119779848793, −9.14098433011956119650001276334, −8.13120373553308838293196644488, −7.44087820432087110207113771052, −6.73172859948171226455069514515, −5.22435834943538864572503031706, −4.41358302594484093101018445672, −3.66354354277304094068879133925, −2.3805995568264001910434748246, −1.273789932029534632028460859072, −0.42101137277470813105415786030, 0.44090354468555564724786757531, 1.96672022507267889653536296626, 3.85460276811644331942320493577, 4.49336896684438255580237811418, 5.069782441013986092746312841783, 6.3062605906781266609790808792, 6.75934163362149796040699268958, 8.01146336631308095613332615108, 8.72123418314197319232992525918, 9.4513340684590664663856469573, 10.42733549275681391250067881037, 11.40641196456101117427547660053, 12.10902686431778259946100854098, 12.82965167499625595869220264296, 14.422685605395002205661512992708, 15.02006795396556890545174356755, 15.34722461456487142421874991969, 16.35097045155381213617549671387, 16.97699437541042712743430668913, 17.59226354544465306627999694668, 18.47973698928401514812238925320, 19.33162473772346132162906846645, 20.04307485283448004733207965224, 21.32020193530745162091444633536, 22.103981300645366526002549422313

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