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Properties

Label 1-967-967.128-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0416 + 0.999i$
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  + (−0.829 − 0.557i)2-s + (−0.334 + 0.942i)3-s + (0.377 + 0.926i)4-s + (−0.247 − 0.968i)5-s + (0.803 − 0.595i)6-s + (−0.877 − 0.480i)7-s + (0.203 − 0.979i)8-s + (−0.775 − 0.631i)9-s + (−0.334 + 0.942i)10-s + (−0.775 − 0.631i)11-s + (−0.998 + 0.0455i)12-s + (0.934 + 0.356i)13-s + (0.460 + 0.887i)14-s + (0.995 + 0.0909i)15-s + (−0.715 + 0.699i)16-s + (−0.576 − 0.816i)17-s + ⋯
L(s)  = 1  + (−0.829 − 0.557i)2-s + (−0.334 + 0.942i)3-s + (0.377 + 0.926i)4-s + (−0.247 − 0.968i)5-s + (0.803 − 0.595i)6-s + (−0.877 − 0.480i)7-s + (0.203 − 0.979i)8-s + (−0.775 − 0.631i)9-s + (−0.334 + 0.942i)10-s + (−0.775 − 0.631i)11-s + (−0.998 + 0.0455i)12-s + (0.934 + 0.356i)13-s + (0.460 + 0.887i)14-s + (0.995 + 0.0909i)15-s + (−0.715 + 0.699i)16-s + (−0.576 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0416 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0416 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0416 + 0.999i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0416 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09425319897 + 0.09040208862i\)
\(L(\frac12)\) \(\approx\) \(0.09425319897 + 0.09040208862i\)
\(L(1)\) \(\approx\) \(0.4223196786 - 0.08947031232i\)
\(L(1)\) \(\approx\) \(0.4223196786 - 0.08947031232i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.829 - 0.557i)T \)
3 \( 1 + (-0.334 + 0.942i)T \)
5 \( 1 + (-0.247 - 0.968i)T \)
7 \( 1 + (-0.877 - 0.480i)T \)
11 \( 1 + (-0.775 - 0.631i)T \)
13 \( 1 + (0.934 + 0.356i)T \)
17 \( 1 + (-0.576 - 0.816i)T \)
19 \( 1 + (-0.648 - 0.761i)T \)
23 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (-0.775 + 0.631i)T \)
31 \( 1 + (-0.949 + 0.313i)T \)
37 \( 1 + (0.934 + 0.356i)T \)
41 \( 1 + (-0.917 - 0.398i)T \)
43 \( 1 + (-0.419 - 0.907i)T \)
47 \( 1 + (0.995 + 0.0909i)T \)
53 \( 1 + (0.613 - 0.789i)T \)
59 \( 1 + (0.613 + 0.789i)T \)
61 \( 1 + (0.113 + 0.993i)T \)
67 \( 1 + (0.203 + 0.979i)T \)
71 \( 1 + (-0.0682 - 0.997i)T \)
73 \( 1 + (0.613 + 0.789i)T \)
79 \( 1 + (0.613 + 0.789i)T \)
83 \( 1 + (0.803 - 0.595i)T \)
89 \( 1 + (0.746 - 0.665i)T \)
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

−21.790924629324090081428983629966, −20.335893069558108178251048702265, −19.69668237581024455335493309196, −18.80856077250274224525097483847, −18.45293667063101675056413219580, −17.90220141916652880704473024902, −16.94993376058541967998566373932, −16.06850833457522788079075840402, −15.30758700731697928872945143489, −14.683141879721848094207251917642, −13.52919299529624101992693605427, −12.83326728746933280059726312253, −11.81247106737527486727195807037, −10.87314153536875257029855154515, −10.354920297244709887768393518258, −9.32604520569635932059538060395, −8.09608088383895933416236158400, −7.7814679564822414669984101059, −6.62893482692283087120728103644, −6.21055774969848155583430902458, −5.516983996266160250467045370441, −3.7889340431495556363625571870, −2.45310112750175763533925231392, −1.85350431115550686670499555791, −0.10510136251290761568895818202, 0.799868653508714259133554521221, 2.3857054528457688295677736793, 3.57631247323377707774501499329, 4.06886793517782576258036027625, 5.21842591509364393718460703664, 6.29275162079586057724946349755, 7.336789607715772749273303098193, 8.639400539240750133591840963489, 8.89194450847570262833065642475, 9.819604996610122742041423822998, 10.62940542789287005727394026878, 11.27012837982791732706912593642, 12.08414616339259757688064588604, 13.12316640673787807827904777086, 13.59802442857906276560818991887, 15.301715172564652543588988320669, 16.172933147051139919590810567398, 16.24756888323026573370740811861, 17.04227802175786496837474209030, 18.01582602062982335926396260655, 18.80748057414625185410398865462, 19.924105187376990817305213082413, 20.25778928704260371243325444686, 20.9717643607365647929587475359, 21.765996848547861331296326236111

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