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Properties

Label 1-731-731.67-r0-0-0
Degree $1$
Conductor $731$
Sign $0.953 - 0.300i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (−0.222 − 0.974i)2-s + (−0.955 + 0.294i)3-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + (0.623 + 0.781i)8-s + (0.826 − 0.563i)9-s + (0.988 + 0.149i)10-s + (0.900 + 0.433i)11-s + (0.733 − 0.680i)12-s + (−0.988 + 0.149i)13-s + (−0.955 − 0.294i)14-s + (0.0747 − 0.997i)15-s + (0.623 − 0.781i)16-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.955 + 0.294i)3-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + (0.623 + 0.781i)8-s + (0.826 − 0.563i)9-s + (0.988 + 0.149i)10-s + (0.900 + 0.433i)11-s + (0.733 − 0.680i)12-s + (−0.988 + 0.149i)13-s + (−0.955 − 0.294i)14-s + (0.0747 − 0.997i)15-s + (0.623 − 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.953 - 0.300i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.953 - 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7680599767 - 0.1180430306i\)
\(L(\frac12)\) \(\approx\) \(0.7680599767 - 0.1180430306i\)
\(L(1)\) \(\approx\) \(0.6660637951 - 0.1515523590i\)
\(L(1)\) \(\approx\) \(0.6660637951 - 0.1515523590i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (-0.955 + 0.294i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (0.826 + 0.563i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + (0.733 - 0.680i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.988 - 0.149i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (0.826 + 0.563i)T \)
71 \( 1 + (-0.0747 + 0.997i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.955 - 0.294i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
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

−22.61910779508497075852354382668, −22.02797859977313268461480295460, −21.27341438250413623064648776492, −19.75444215370488764359875304164, −19.226243156863298652000603044597, −18.19475883322053812408686182684, −17.472441126175165509441987034482, −16.96234192902254342978964432160, −16.0748045469904919075514187398, −15.51557254269477418604836674320, −14.5168348057805940717053917955, −13.51371658710296542849127049492, −12.55598526395647168625333868315, −11.90042169353919216758347076635, −11.09515753422573477229482555552, −9.629047853191897056722686215809, −9.09749515841111034578022930264, −8.01402638784704036273661815358, −7.331399390754534939362692172816, −6.28343641919014799655762972971, −5.23147705362081618489192274154, −5.0712845771351246801226540933, −3.81908466857698088899923926094, −1.763711557736854508749598910857, −0.7022341409874519670766585981, 0.8352437085290645659384043661, 2.04076701754417362755303547228, 3.401713476952975979188720631850, 4.24311116434791443386566103357, 4.86914445670941578738771060020, 6.32799052584733208780858540056, 7.25899935778041615539198730485, 8.03216896107739948107625262712, 9.76918038225927389482537870390, 9.890899108002325168068357524898, 10.99400877057226119360864528659, 11.54781473857094368756786334899, 12.124987525528276572591473492746, 13.20204714830532946671977166264, 14.4240632463762395140322375285, 14.77524547816444900136421432460, 16.27657888168386660082079026610, 17.08416650221709232029452205456, 17.61111575797479397641663471356, 18.46200565154010464884486281694, 19.188187236353550736837796744954, 20.166413160543346492203515614425, 20.78017511950052372423248067210, 21.88760763645283272942676347711, 22.44058177371618574283811375697

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