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Properties

Label 1-967-967.143-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.887 + 0.461i$
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  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + 9-s − 10-s + 11-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s − 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + 9-s − 10-s + 11-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s − 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.887 + 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2320953496 - 0.9486962618i\)
\(L(\frac12)\) \(\approx\) \(-0.2320953496 - 0.9486962618i\)
\(L(1)\) \(\approx\) \(0.5465317932 - 0.4817018143i\)
\(L(1)\) \(\approx\) \(0.5465317932 - 0.4817018143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−22.09864863232539975849761770136, −21.70198265286747837795236066386, −20.45516719137109313030515777951, −19.01886026357008336844133164585, −18.643649863003707528812261408669, −18.065301979568207990250219206414, −17.20935372828164139978028358289, −16.7229075174429175338092784778, −15.710078359027202839974506096831, −15.04398598217677668269382250874, −14.22197320163318073390667230145, −13.54915192510403611756370568806, −12.16987111087383783703326611313, −11.39275657395002403395089612503, −10.7842100593459343001656500552, −9.56362740573478789100639516094, −9.311481992880764547562232156036, −7.91971706882979359226428346250, −7.0761969280444675882820467490, −6.22346040226235885704726524556, −5.82516084660661088917862383956, −4.8691884215044092316701858024, −3.77271220477824975145262859655, −2.01445267354486522609832450373, −1.233501318593550731358068443119, 0.31874527751398656794771145003, 1.27201346140197100237754500995, 1.63095106227476025779655243866, 3.55970552379846443869038500452, 4.1868192372237016561770152231, 5.225409580234309797542753000509, 5.991770883967339833652384932383, 7.35820511047152584925235523500, 8.0405694310451387133045141711, 9.12667520202771896495302795300, 10.03463573913814003684787509832, 10.472010929811399617295486569154, 11.503277551695511690411735490889, 12.119644902868919916022647043682, 12.83530574957713977030263353624, 13.63902605027369870817624458308, 14.505995118709608070015618432807, 16.20343841863942064423555340126, 16.57197549466807434469928931595, 17.26974043685132866676265824093, 17.90075331453564180627893554972, 18.51453967908596722852311930270, 19.75912243742925958173607262967, 20.339408847924206077532955190487, 21.0349396954560266786383460589

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