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Properties

Label 1-731-731.260-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.999 - 0.0369i$
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  + (−0.111 − 0.993i)2-s + (0.960 − 0.276i)3-s + (−0.974 + 0.222i)4-s + (−0.666 − 0.745i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)7-s + (0.330 + 0.943i)8-s + (0.846 − 0.532i)9-s + (−0.666 + 0.745i)10-s + (−0.985 + 0.167i)11-s + (−0.875 + 0.483i)12-s + (0.433 − 0.900i)13-s + (−0.483 − 0.875i)14-s + (−0.846 − 0.532i)15-s + (0.900 − 0.433i)16-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)2-s + (0.960 − 0.276i)3-s + (−0.974 + 0.222i)4-s + (−0.666 − 0.745i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)7-s + (0.330 + 0.943i)8-s + (0.846 − 0.532i)9-s + (−0.666 + 0.745i)10-s + (−0.985 + 0.167i)11-s + (−0.875 + 0.483i)12-s + (0.433 − 0.900i)13-s + (−0.483 − 0.875i)14-s + (−0.846 − 0.532i)15-s + (0.900 − 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.999 - 0.0369i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (260, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.999 - 0.0369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02647860826 - 1.433386921i\)
\(L(\frac12)\) \(\approx\) \(0.02647860826 - 1.433386921i\)
\(L(1)\) \(\approx\) \(0.7479370395 - 0.8799052029i\)
\(L(1)\) \(\approx\) \(0.7479370395 - 0.8799052029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.111 - 0.993i)T \)
3 \( 1 + (0.960 - 0.276i)T \)
5 \( 1 + (-0.666 - 0.745i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (-0.985 + 0.167i)T \)
13 \( 1 + (0.433 - 0.900i)T \)
19 \( 1 + (-0.846 - 0.532i)T \)
23 \( 1 + (0.985 - 0.167i)T \)
29 \( 1 + (-0.875 + 0.483i)T \)
31 \( 1 + (-0.483 - 0.875i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (-0.276 + 0.960i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.330 - 0.943i)T \)
59 \( 1 + (-0.943 - 0.330i)T \)
61 \( 1 + (-0.875 - 0.483i)T \)
67 \( 1 + (0.222 + 0.974i)T \)
71 \( 1 + (-0.985 - 0.167i)T \)
73 \( 1 + (-0.666 - 0.745i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.993 + 0.111i)T \)
89 \( 1 + (-0.781 + 0.623i)T \)
97 \( 1 + (-0.578 + 0.815i)T \)
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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

−23.2841286220290842184364598456, −22.05768305439953152752771168515, −21.37114598166323197643451398521, −20.62872210389009433891067863461, −19.35487351255379586162502524512, −18.63117688474029716244317005574, −18.42061378461041861450463845445, −17.0886319209752820061910798743, −16.1475706310397307853585791192, −15.28376325055236413517057436050, −14.99672881064484337159101651092, −14.126627610753670323459783922981, −13.48375504990046837504284741785, −12.34519099072956924603410575522, −10.99484307840857499759013723392, −10.3807902208866800374383259094, −9.086464485725292397157833285441, −8.50691936703723361963886235824, −7.700067143259423825800654507690, −7.12990675846857603954977010936, −5.89876494516878122035503635162, −4.73891760136338728364388681243, −4.03100751062384927712253294988, −2.94719443454683030176082439137, −1.696065381667486690167829098098, 0.64870296103276296366518148615, 1.69848533540627423022712245940, 2.708524926224095868620389564102, 3.73476655920935869586330468666, 4.53095622141313798927079500198, 5.37513526067881984841020736787, 7.38055166948533993011297628400, 7.980380580432472179213833625490, 8.61192495760052297258103701841, 9.40264732841008975843147999498, 10.63325484013213316606531540144, 11.155512746385250681398977967150, 12.37316530035222866423574703916, 13.06428919147639229324834778932, 13.45270288325821867784088057038, 14.78837704983625169456694244893, 15.21919898296760098326026341489, 16.52514111036434852786670650139, 17.55027740156874262023611240248, 18.304674366632086425371004059291, 19.037385852231662515612385907036, 19.93578656162932560194102025833, 20.51463332223051279227132326402, 20.876896413300999464517425259441, 21.725341847842155316102475677317

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