Properties

Label 1-967-967.168-r0-0-0
Degree $1$
Conductor $967$
Sign $0.999 + 0.0224i$
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  + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (−0.988 + 0.149i)5-s + (0.955 − 0.294i)6-s + (0.826 − 0.563i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.955 + 0.294i)12-s + (−0.988 + 0.149i)13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.733 + 0.680i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (−0.988 + 0.149i)5-s + (0.955 − 0.294i)6-s + (0.826 − 0.563i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.955 + 0.294i)12-s + (−0.988 + 0.149i)13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.733 + 0.680i)16-s + (0.623 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.999 + 0.0224i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.999 + 0.0224i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.848634210 + 0.03196516127i\)
\(L(\frac12)\) \(\approx\) \(2.848634210 + 0.03196516127i\)
\(L(1)\) \(\approx\) \(1.918390828 + 0.1259884996i\)
\(L(1)\) \(\approx\) \(1.918390828 + 0.1259884996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.826 + 0.563i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + (0.826 + 0.563i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.955 + 0.294i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−21.85518863818265569868965786307, −20.791671841730483843483456965875, −20.32312454494505127812267293009, −19.74254693708621625073514691845, −18.96993225384849182861783618758, −18.03041232193137752006255221412, −16.697067914405329229039293191887, −15.899068322802635917641993803986, −15.12425080536891757598986720660, −14.60990817249708470337073726020, −14.1428711338602694041820201652, −12.76599313675937707990549979360, −12.10820688928783869962647370971, −11.393793709690999560322969619609, −10.67341352314602697450993938544, −9.53142062022568265189687357964, −9.04681464808629188210725827391, −7.74718184842093495735388845432, −7.13650760389858336350892930144, −5.516241264493976429114622568554, −4.62824715778116946890836121565, −4.45054547682471878412414273148, −3.09073184089990267993597063940, −2.57858803877533135125689848557, −1.22098247817729094382039222928, 1.05154337679256717174987205668, 2.34349324209777899315197216761, 3.538897167631391068016561178840, 3.89614904379546328086848188652, 5.12059948907092359948052592303, 6.15167228073085431342085753178, 7.23349563944970742720943620944, 7.63021456132817517590373267419, 8.25254281903578236537014629773, 9.2357904597650223788001731345, 10.82087855565023851611290367780, 11.73008452579732385758276309223, 12.13448732223732271360371640762, 13.09283754929351715918950470857, 14.05797634841343009881202665801, 14.50247344196265764867397576482, 15.05264863328257935848007611672, 16.12692297911136355154664093940, 16.986559385924141261750954593977, 17.63291331333998679056387653329, 18.75705749373016828980282724079, 19.54485609792935342359698813669, 20.12178789789654090042489822290, 21.00808311986925458948447954152, 21.77883885440153341928113460006

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