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Properties

Label 1-967-967.256-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.381 + 0.924i$
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  + (−0.783 + 0.620i)2-s + (0.924 + 0.380i)3-s + (0.228 − 0.973i)4-s + (0.818 + 0.574i)5-s + (−0.961 + 0.276i)6-s + (−0.992 − 0.123i)7-s + (0.425 + 0.905i)8-s + (0.710 + 0.703i)9-s + (−0.998 + 0.0585i)10-s + (−0.844 + 0.536i)11-s + (0.581 − 0.813i)12-s + (0.886 − 0.462i)13-s + (0.854 − 0.519i)14-s + (0.538 + 0.842i)15-s + (−0.895 − 0.445i)16-s + (−0.967 + 0.250i)17-s + ⋯
L(s)  = 1  + (−0.783 + 0.620i)2-s + (0.924 + 0.380i)3-s + (0.228 − 0.973i)4-s + (0.818 + 0.574i)5-s + (−0.961 + 0.276i)6-s + (−0.992 − 0.123i)7-s + (0.425 + 0.905i)8-s + (0.710 + 0.703i)9-s + (−0.998 + 0.0585i)10-s + (−0.844 + 0.536i)11-s + (0.581 − 0.813i)12-s + (0.886 − 0.462i)13-s + (0.854 − 0.519i)14-s + (0.538 + 0.842i)15-s + (−0.895 − 0.445i)16-s + (−0.967 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.381 + 0.924i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.381 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7994989238 + 1.195490703i\)
\(L(\frac12)\) \(\approx\) \(0.7994989238 + 1.195490703i\)
\(L(1)\) \(\approx\) \(0.9080262953 + 0.5554916202i\)
\(L(1)\) \(\approx\) \(0.9080262953 + 0.5554916202i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.783 + 0.620i)T \)
3 \( 1 + (0.924 + 0.380i)T \)
5 \( 1 + (0.818 + 0.574i)T \)
7 \( 1 + (-0.992 - 0.123i)T \)
11 \( 1 + (-0.844 + 0.536i)T \)
13 \( 1 + (0.886 - 0.462i)T \)
17 \( 1 + (-0.967 + 0.250i)T \)
19 \( 1 + (0.692 - 0.721i)T \)
23 \( 1 + (0.987 + 0.155i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (0.139 + 0.990i)T \)
37 \( 1 + (0.190 - 0.981i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (0.177 - 0.984i)T \)
47 \( 1 + (-0.322 + 0.946i)T \)
53 \( 1 + (-0.829 + 0.557i)T \)
59 \( 1 + (-0.953 + 0.300i)T \)
61 \( 1 + (-0.949 - 0.313i)T \)
67 \( 1 + (0.787 + 0.615i)T \)
71 \( 1 + (-0.145 + 0.989i)T \)
73 \( 1 + (-0.829 - 0.557i)T \)
79 \( 1 + (0.989 + 0.142i)T \)
83 \( 1 + (0.483 + 0.875i)T \)
89 \( 1 + (-0.927 + 0.374i)T \)
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

−21.12933309741841298683455176627, −20.66885082325121395224716883356, −19.95765980264503735129808891781, −19.069875702865228115821371055281, −18.52073698612514190127702319492, −17.91008683922383734275179859730, −16.805759675276602056887767848878, −16.08115305943017720007826304637, −15.456448963606083026369238931744, −13.86543485333773886190017906638, −13.38295226130284170663083407827, −12.87744514359896497760243303168, −11.98543638279928092885475542909, −10.819667957691678235835600740938, −9.905861056168051913109770996885, −9.294251416788900687453023838848, −8.66349879809768486616817858635, −7.941870981964567433222625818582, −6.775345842416341484087076491643, −6.09628248676888004032698722176, −4.55323797445135269367100156897, −3.35875386123453350727205191275, −2.72356659853432249000725341706, −1.7913787486472884025222541858, −0.755384804624970907228515146879, 1.31727048096118670032907994056, 2.55997188101083797296607943506, 3.08795553236390454991588036620, 4.61595690977058271842781476285, 5.60252732520539746652677646328, 6.64050685079321981809257770506, 7.195391479186354279371158864578, 8.22406987187344748078243269680, 9.19473060198134554446454476077, 9.56147645525374694347601593108, 10.66406021556958041369189597580, 10.7623258640938300519658491665, 12.809732624185098562915721142409, 13.48841775733771323816110690067, 14.08280195217166186991935496494, 15.17208013470095252759410137992, 15.641632697436534310785835835077, 16.22749847080503816638213222804, 17.45663523779955448728927258534, 18.04299711118860691986887891649, 18.78602584137845397189734721079, 19.60201020165465105649294833689, 20.21637324473903334458790101822, 21.05683357984889668305817359313, 21.94855463512226861767860093460

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