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Properties

Label 1-731-731.135-r0-0-0
Degree $1$
Conductor $731$
Sign $0.216 + 0.976i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s − 11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + ⋯
L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s − 11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.740246877 + 2.198113310i\)
\(L(\frac12)\) \(\approx\) \(2.740246877 + 2.198113310i\)
\(L(1)\) \(\approx\) \(2.184378318 + 0.9447681328i\)
\(L(1)\) \(\approx\) \(2.184378318 + 0.9447681328i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−22.34333632772563208120470744197, −21.40625643083839135784625066708, −20.70158908821460234372613187544, −20.30466208148683440634070776016, −19.20639989097039054905080336405, −18.35799223839107965343104601014, −17.49774932357670883257078942340, −16.5720861136929237955160937299, −15.491951599935845823142450524248, −14.82728823776527374533103290871, −14.03536137635681392973825738145, −13.11137222383855903564746103176, −12.558588204862831485256589954475, −12.13447151034888195082529088916, −10.88521022307277115329545765932, −9.812820651501056914383865011142, −8.45074857336117007340677213712, −8.081970129868779976396033860757, −6.94060077537956222252514655831, −5.733047977035027773925003101603, −5.40214208227151200321096704982, −4.24852297063241003570138917164, −2.72520190068124118273990274987, −2.32878441488320347521403654151, −1.16217994611706309833565913123, 1.89715978503679774049815294338, 2.72031312280374578748861395165, 3.56628393349480104257786692112, 4.63757457027172119975427358892, 5.15004018502120758993710328751, 6.4725603101904963182163356962, 7.28782860029657450003145963004, 8.158715458003901753418567500252, 9.60780188656311221677668407909, 10.3925832812981599625665309532, 10.95771769934282316191954862357, 11.744750064997433656493707084745, 13.37663149830502240328098968327, 13.633143291876134856394332668657, 14.4819739227161791726895008780, 15.144354224525172002143118060227, 15.82575659400755741272255062745, 16.90028390781336098480843643236, 17.5304394404620305516680576273, 18.979819354742256014983049835658, 19.660565901922526016817613637241, 20.61754146292349851664865321237, 21.317896645503562330991366101633, 21.65726837442247778019762221250, 22.58038760874392736349605000110

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