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Properties

Label 1-967-967.615-r1-0-0
Degree $1$
Conductor $967$
Sign $0.981 - 0.190i$
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  + (0.291 − 0.956i)2-s + (0.576 + 0.816i)3-s + (−0.829 − 0.557i)4-s + (−0.613 + 0.789i)5-s + (0.949 − 0.313i)6-s + (0.247 − 0.968i)7-s + (−0.775 + 0.631i)8-s + (−0.334 + 0.942i)9-s + (0.576 + 0.816i)10-s + (−0.334 + 0.942i)11-s + (−0.0227 − 0.999i)12-s + (−0.983 − 0.181i)13-s + (−0.854 − 0.519i)14-s + (−0.998 − 0.0455i)15-s + (0.377 + 0.926i)16-s + (0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (0.291 − 0.956i)2-s + (0.576 + 0.816i)3-s + (−0.829 − 0.557i)4-s + (−0.613 + 0.789i)5-s + (0.949 − 0.313i)6-s + (0.247 − 0.968i)7-s + (−0.775 + 0.631i)8-s + (−0.334 + 0.942i)9-s + (0.576 + 0.816i)10-s + (−0.334 + 0.942i)11-s + (−0.0227 − 0.999i)12-s + (−0.983 − 0.181i)13-s + (−0.854 − 0.519i)14-s + (−0.998 − 0.0455i)15-s + (0.377 + 0.926i)16-s + (0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.981 - 0.190i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.868526057 - 0.1795037992i\)
\(L(\frac12)\) \(\approx\) \(1.868526057 - 0.1795037992i\)
\(L(1)\) \(\approx\) \(1.112242880 - 0.2049000890i\)
\(L(1)\) \(\approx\) \(1.112242880 - 0.2049000890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.291 - 0.956i)T \)
3 \( 1 + (0.576 + 0.816i)T \)
5 \( 1 + (-0.613 + 0.789i)T \)
7 \( 1 + (0.247 - 0.968i)T \)
11 \( 1 + (-0.334 + 0.942i)T \)
13 \( 1 + (-0.983 - 0.181i)T \)
17 \( 1 + (0.460 - 0.887i)T \)
19 \( 1 + (0.419 - 0.907i)T \)
23 \( 1 + (0.0682 - 0.997i)T \)
29 \( 1 + (0.334 + 0.942i)T \)
31 \( 1 + (-0.158 - 0.987i)T \)
37 \( 1 + (-0.983 - 0.181i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (-0.538 + 0.842i)T \)
47 \( 1 + (0.998 + 0.0455i)T \)
53 \( 1 + (0.898 - 0.439i)T \)
59 \( 1 + (0.898 + 0.439i)T \)
61 \( 1 + (0.746 + 0.665i)T \)
67 \( 1 + (0.775 + 0.631i)T \)
71 \( 1 + (0.682 - 0.730i)T \)
73 \( 1 + (0.898 + 0.439i)T \)
79 \( 1 + (-0.898 - 0.439i)T \)
83 \( 1 + (-0.949 + 0.313i)T \)
89 \( 1 + (-0.934 + 0.356i)T \)
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

−21.46932687233988938810556651250, −21.04512682020967330721008742065, −19.80887703116145919112415564800, −18.97315107340713471569917773173, −18.633687492544556406670642278748, −17.41499303661660063839425070556, −16.94134427432524558203149523052, −15.75246905175755422549367767672, −15.37122671491879045048962254665, −14.36991435144318346960444080029, −13.79038866085387742921121753461, −12.76204724318601951554324442463, −12.23627043530286283099308456764, −11.67573165171956712088384490563, −9.87176460157522063142202534265, −8.85270324292334283248385934017, −8.37917495556847814391764669714, −7.77784015827281694015579140272, −6.91245070931573295711882593121, −5.64646482354447103928733406120, −5.354696798911074945946693840838, −3.91207349220420033732629581064, −3.18089655721207595391381813318, −1.84608739756495389797085672577, −0.54065553690096963474948431837, 0.63205093252747994625457156447, 2.29426115965595234774393756894, 2.88532973002255414495687793820, 3.81318518874726057770698113911, 4.63053453233887426947841048567, 5.14628568473532255272778095996, 6.966547531588853933430071977656, 7.62259497949284479154101611729, 8.66769500836177065350936983526, 9.94147000666068340197974841162, 10.047940988033376103441999220811, 11.0150214386925045655742114563, 11.66502896186059217758925432624, 12.71893102062572240682350208420, 13.679934378772809116563902618372, 14.47574501128632427620336274328, 14.83623270218612944483398559064, 15.74573067231718650554433335844, 16.80939509040391083070188780142, 17.82147920863224378449879857031, 18.565296411789031142709080783661, 19.61779230846842734213128195874, 20.01592354185733651913413141405, 20.590032370089610985902392039042, 21.4242425964404567095980306591

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