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Properties

Label 1-731-731.3-r0-0-0
Degree $1$
Conductor $731$
Sign $0.627 + 0.778i$
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  + (−0.993 − 0.111i)2-s + (0.856 + 0.516i)3-s + (0.974 + 0.222i)4-s + (0.836 − 0.547i)5-s + (−0.793 − 0.608i)6-s + (−0.991 − 0.130i)7-s + (−0.943 − 0.330i)8-s + (0.467 + 0.884i)9-s + (−0.892 + 0.450i)10-s + (0.578 + 0.815i)11-s + (0.720 + 0.693i)12-s + (0.997 + 0.0747i)13-s + (0.970 + 0.240i)14-s + (0.999 − 0.0373i)15-s + (0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (−0.993 − 0.111i)2-s + (0.856 + 0.516i)3-s + (0.974 + 0.222i)4-s + (0.836 − 0.547i)5-s + (−0.793 − 0.608i)6-s + (−0.991 − 0.130i)7-s + (−0.943 − 0.330i)8-s + (0.467 + 0.884i)9-s + (−0.892 + 0.450i)10-s + (0.578 + 0.815i)11-s + (0.720 + 0.693i)12-s + (0.997 + 0.0747i)13-s + (0.970 + 0.240i)14-s + (0.999 − 0.0373i)15-s + (0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243171208 + 0.5945202727i\)
\(L(\frac12)\) \(\approx\) \(1.243171208 + 0.5945202727i\)
\(L(1)\) \(\approx\) \(1.020445344 + 0.1974856622i\)
\(L(1)\) \(\approx\) \(1.020445344 + 0.1974856622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.993 - 0.111i)T \)
3 \( 1 + (0.856 + 0.516i)T \)
5 \( 1 + (0.836 - 0.547i)T \)
7 \( 1 + (-0.991 - 0.130i)T \)
11 \( 1 + (0.578 + 0.815i)T \)
13 \( 1 + (0.997 + 0.0747i)T \)
19 \( 1 + (-0.467 + 0.884i)T \)
23 \( 1 + (-0.416 + 0.908i)T \)
29 \( 1 + (0.240 - 0.970i)T \)
31 \( 1 + (-0.693 + 0.720i)T \)
37 \( 1 + (0.130 + 0.991i)T \)
41 \( 1 + (0.483 - 0.875i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (0.757 + 0.652i)T \)
59 \( 1 + (0.330 + 0.943i)T \)
61 \( 1 + (0.720 - 0.693i)T \)
67 \( 1 + (-0.955 + 0.294i)T \)
71 \( 1 + (0.416 + 0.908i)T \)
73 \( 1 + (-0.892 - 0.450i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
83 \( 1 + (0.804 - 0.593i)T \)
89 \( 1 + (0.149 - 0.988i)T \)
97 \( 1 + (-0.985 - 0.167i)T \)
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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

−22.23471059051621811346275827164, −21.406063478847496458871259869854, −20.60185230620496690782139049362, −19.62395632421786599101181260808, −19.21110960240396211470555107233, −18.26653004634666884912927004483, −17.964539679089610245107677417186, −16.68669109285617404138450978815, −16.05894090509775178584517888124, −14.99302324528195663051522304413, −14.30080479505313112128138839763, −13.34518209533530829480715322450, −12.6571862298888651786016961950, −11.3481466393743928355126454981, −10.540904290247921016594848766823, −9.548460663366976572373280088164, −8.999604767121901675528888954028, −8.28657850187877856734777028050, −7.01145711480790205065374497766, −6.46763248252050010836350381625, −5.849008347848573253380973445706, −3.66083861338743006799167052902, −2.873661598370549905967105028135, −2.038311859680966767863124177988, −0.85665874746842795455623922283, 1.37742953930448069787732847037, 2.14501625554794572468114011179, 3.32490200380461428121332131575, 4.17154548628101519924134640694, 5.73389421403827341556508656568, 6.56899039579351934893465505769, 7.625186698416401231538482702973, 8.68017717631445724245184145332, 9.19194526659751398479154350932, 10.01454300056315685019350442358, 10.368796359172053016866067653138, 11.79673886622829128026826518870, 12.80162781524112302367160651692, 13.519347194310283119982025128776, 14.54375142751560662004699765242, 15.5477535653957122431096513528, 16.21271217122146505723472219844, 16.877540767143307717822262794897, 17.749038853601796778691646478501, 18.70953187301447802367343643942, 19.50592806362492882336421469279, 20.1678194579567577645977044179, 20.817910193783113849587090436046, 21.47050103763139056155124997865, 22.381189364636220079522945740526

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