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Properties

Label 1-967-967.67-r1-0-0
Degree $1$
Conductor $967$
Sign $0.982 - 0.183i$
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  + (0.763 − 0.646i)2-s + (0.297 − 0.954i)3-s + (0.165 − 0.986i)4-s + (0.107 − 0.994i)5-s + (−0.389 − 0.921i)6-s + (−0.787 + 0.615i)7-s + (−0.511 − 0.859i)8-s + (−0.822 − 0.568i)9-s + (−0.560 − 0.828i)10-s + (0.494 + 0.869i)11-s + (−0.892 − 0.451i)12-s + (−0.494 + 0.869i)13-s + (−0.203 + 0.979i)14-s + (−0.917 − 0.398i)15-s + (−0.945 − 0.325i)16-s + (0.999 + 0.0390i)17-s + ⋯
L(s)  = 1  + (0.763 − 0.646i)2-s + (0.297 − 0.954i)3-s + (0.165 − 0.986i)4-s + (0.107 − 0.994i)5-s + (−0.389 − 0.921i)6-s + (−0.787 + 0.615i)7-s + (−0.511 − 0.859i)8-s + (−0.822 − 0.568i)9-s + (−0.560 − 0.828i)10-s + (0.494 + 0.869i)11-s + (−0.892 − 0.451i)12-s + (−0.494 + 0.869i)13-s + (−0.203 + 0.979i)14-s + (−0.917 − 0.398i)15-s + (−0.945 − 0.325i)16-s + (0.999 + 0.0390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.982 - 0.183i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.982 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.820962021 - 0.1688445688i\)
\(L(\frac12)\) \(\approx\) \(1.820962021 - 0.1688445688i\)
\(L(1)\) \(\approx\) \(1.144892387 - 0.8579317323i\)
\(L(1)\) \(\approx\) \(1.144892387 - 0.8579317323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.763 - 0.646i)T \)
3 \( 1 + (0.297 - 0.954i)T \)
5 \( 1 + (0.107 - 0.994i)T \)
7 \( 1 + (-0.787 + 0.615i)T \)
11 \( 1 + (0.494 + 0.869i)T \)
13 \( 1 + (-0.494 + 0.869i)T \)
17 \( 1 + (0.999 + 0.0390i)T \)
19 \( 1 + (0.864 + 0.502i)T \)
23 \( 1 + (-0.874 + 0.485i)T \)
29 \( 1 + (0.371 + 0.928i)T \)
31 \( 1 + (0.833 + 0.552i)T \)
37 \( 1 + (0.957 + 0.288i)T \)
41 \( 1 + (-0.962 - 0.269i)T \)
43 \( 1 + (-0.653 - 0.756i)T \)
47 \( 1 + (-0.592 + 0.805i)T \)
53 \( 1 + (-0.576 + 0.816i)T \)
59 \( 1 + (-0.668 + 0.744i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (-0.833 + 0.552i)T \)
71 \( 1 + (0.763 + 0.646i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (0.998 + 0.0585i)T \)
83 \( 1 + (-0.750 - 0.660i)T \)
89 \( 1 + (-0.833 + 0.552i)T \)
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

−21.82875662056556762925209871961, −21.07029770591116060308533555381, −20.0785270767589840083097193666, −19.49370295248298013234889752844, −18.33575608623321991221196679214, −17.26477139647066343869889736477, −16.638233888445648105888591517936, −15.90394788573922708149360116049, −15.21673014137308947031842017488, −14.401833694328285663159885556061, −13.893826680227484036643485256546, −13.175812863772836870360037549419, −11.86393692957819816933895933796, −11.19534602429945570343325418776, −10.06845458426025232271566399804, −9.679803283730640657497071428879, −8.23149652602493227865776128365, −7.661362498274811354205915196655, −6.497073963084035860861720617238, −5.92043558222043001615053800741, −4.89839729747296183194670747487, −3.77745215827398666603352812317, −3.25021936720528543761133332923, −2.61345555367216443131600796158, −0.27436787782841346404096556003, 1.183931796243674395265451229605, 1.75192234115977535835942990662, 2.815780513973599193941478815166, 3.74878187483006839673069220753, 4.87238366566445404763347277481, 5.73419306957297970783830713805, 6.51255271572730881387612211981, 7.44408764244860064455353265667, 8.65011544180668349726067120001, 9.53790779527779818253237492983, 9.949224166115098727958093795425, 11.76944650131785146035577299086, 12.105400711243980344823759157193, 12.50818482645889173549833006052, 13.47182334653921937081911212275, 14.1049793167481807430917212123, 14.89317273156685329215228247853, 15.93847295304951615910239712394, 16.72953203224242923182758776103, 17.82410294198412360757465937724, 18.67023719139165362015999283273, 19.38375362153198768017115725357, 19.992759368710405495362225811835, 20.56855522771088979258929354275, 21.54498495491698939604741271057

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