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Properties

Label 1-967-967.142-r0-0-0
Degree $1$
Conductor $967$
Sign $0.722 - 0.691i$
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  + (−0.5 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310891783 - 0.5262736139i\)
\(L(\frac12)\) \(\approx\) \(1.310891783 - 0.5262736139i\)
\(L(1)\) \(\approx\) \(1.063613187 + 0.02066633763i\)
\(L(1)\) \(\approx\) \(1.063613187 + 0.02066633763i\)

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

−21.58620238234505709208087372851, −21.20282498725366142320618115706, −19.89434406468067455199810890546, −19.410419497158427071915011532997, −19.00880242878916334092892262577, −18.36135123491948073098235831189, −17.31390788305596082349313446816, −16.30613121847498908042537612827, −15.43091039590634499031886255659, −14.52734012321672907907209884353, −14.022770692093201361290340807939, −12.85348160052576801530819920076, −12.14235496195937116578609518335, −11.46988266481200849981731973338, −10.41935048964108612677789647835, −9.58931864442827882350612058471, −8.97840598007943739542709884772, −8.27218076635421583063086356091, −7.13637766848788945640960635371, −6.65037614245817103850085397614, −4.80504962016265052188397511195, −3.826229509414670407365606955633, −3.04944901235065542165695719722, −2.47752679328347651627155168883, −1.35378975559386546604400533397, 0.7203525755762025761237776613, 1.58145939879692380740025048518, 3.31354732837801744823427727695, 4.09136210266240526672353712624, 4.93805415600295102205248521071, 6.15035263215745795245905218198, 7.1967435656329562336123002966, 7.79705581213809239566535761125, 8.46074210697996035111220957825, 9.417152907132278581044972359676, 9.87019648130835467389294749393, 10.87896217610868620147274442742, 12.43220047807018599303794918025, 12.945800066619373049344259261518, 13.976364517156141182134553535292, 14.58316694520359928370967828628, 15.330499899882157726796118290027, 16.222015487411755716939573080144, 16.80644660798539183795917842266, 17.43760714852231896748617103807, 18.72985194862762991800323470318, 19.38571189650079190096746826277, 19.88648546127961984728516364231, 20.50737536707201095991634722640, 21.528218616139262281595043648357

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