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Properties

Label 1-967-967.175-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.362 - 0.931i$
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  + (−0.222 − 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.900 + 0.433i)5-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.222 − 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.623 + 0.781i)12-s + (0.900 + 0.433i)13-s − 14-s + 15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.900 + 0.433i)5-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.222 − 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.623 + 0.781i)12-s + (0.900 + 0.433i)13-s − 14-s + 15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.362 - 0.931i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.362 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.800387538 - 2.632268286i\)
\(L(\frac12)\) \(\approx\) \(1.800387538 - 2.632268286i\)
\(L(1)\) \(\approx\) \(1.233360663 - 0.8681427872i\)
\(L(1)\) \(\approx\) \(1.233360663 - 0.8681427872i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + (0.222 + 0.974i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 - T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.623 + 0.781i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−21.78792772147581648156527662212, −21.13025957880241420082176353801, −20.2990251197121807498401237458, −19.30421599399767954931784470430, −18.37386961142541430460985036082, −17.99986203783801695173422823946, −16.979504901312621091808184633050, −15.959695391895155847242368392958, −15.58463623459037999103433494299, −14.82944331840639789921153507721, −13.90201699197280125864236529632, −13.24389690372765851554017381882, −12.76536101300021519913662315925, −10.93759117740913286286326238357, −10.27741205745228246820981588801, −9.200755053582148609631101635627, −8.685653166073053672500977908245, −8.31594853630610323579824626475, −7.03850922114762267345751315639, −6.02323140970390421958036826099, −5.198180285748786384487996126954, −4.621753981139055929989801931839, −3.17429679346951281411829932207, −2.23499470136418799161448981447, −1.00867884884057790247692950547, 0.74176400762579791724027151281, 1.74277876948313418162175111524, 2.38054181896989896429004361840, 3.40606140872672167388763542327, 4.1837380759656584471562083982, 5.35861282456649006249936093485, 6.741697560226100900280558332944, 7.493059362680117669201250829988, 8.39916732223468256719953619669, 9.27341096758094265940366007846, 10.00180474074020654969274222975, 10.69277570799655903364084093139, 11.5124968182039165055622024142, 12.8701176891637014167896164619, 13.2429917794112525261084153972, 13.94967844248895188336868017885, 14.48185231100015510097198174576, 15.701809728183503608942608465822, 16.90834890713810040823834675040, 17.77541903418046437266892890821, 18.30852064006061693797364066324, 18.88935521560979147452860654960, 19.92212469016384645204975452559, 20.551015910125611392211726163794, 21.046595248894430129802614896733

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