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Properties

Label 1-731-731.180-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.287 + 0.957i$
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  + (−0.330 − 0.943i)2-s + (0.745 + 0.666i)3-s + (−0.781 + 0.623i)4-s + (0.578 + 0.815i)5-s + (0.382 − 0.923i)6-s + (−0.923 − 0.382i)7-s + (0.846 + 0.532i)8-s + (0.111 + 0.993i)9-s + (0.578 − 0.815i)10-s + (0.483 + 0.875i)11-s + (−0.998 − 0.0560i)12-s + (−0.974 + 0.222i)13-s + (−0.0560 + 0.998i)14-s + (−0.111 + 0.993i)15-s + (0.222 − 0.974i)16-s + ⋯
L(s)  = 1  + (−0.330 − 0.943i)2-s + (0.745 + 0.666i)3-s + (−0.781 + 0.623i)4-s + (0.578 + 0.815i)5-s + (0.382 − 0.923i)6-s + (−0.923 − 0.382i)7-s + (0.846 + 0.532i)8-s + (0.111 + 0.993i)9-s + (0.578 − 0.815i)10-s + (0.483 + 0.875i)11-s + (−0.998 − 0.0560i)12-s + (−0.974 + 0.222i)13-s + (−0.0560 + 0.998i)14-s + (−0.111 + 0.993i)15-s + (0.222 − 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5769092487 + 0.7754694327i\)
\(L(\frac12)\) \(\approx\) \(0.5769092487 + 0.7754694327i\)
\(L(1)\) \(\approx\) \(0.9157953562 + 0.1577673768i\)
\(L(1)\) \(\approx\) \(0.9157953562 + 0.1577673768i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.330 - 0.943i)T \)
3 \( 1 + (0.745 + 0.666i)T \)
5 \( 1 + (0.578 + 0.815i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (0.483 + 0.875i)T \)
13 \( 1 + (-0.974 + 0.222i)T \)
19 \( 1 + (-0.111 + 0.993i)T \)
23 \( 1 + (-0.483 - 0.875i)T \)
29 \( 1 + (-0.998 - 0.0560i)T \)
31 \( 1 + (-0.0560 + 0.998i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.666 - 0.745i)T \)
47 \( 1 + (0.781 - 0.623i)T \)
53 \( 1 + (0.846 - 0.532i)T \)
59 \( 1 + (0.532 + 0.846i)T \)
61 \( 1 + (-0.998 + 0.0560i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (0.483 - 0.875i)T \)
73 \( 1 + (0.578 + 0.815i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (-0.943 - 0.330i)T \)
89 \( 1 + (0.433 + 0.900i)T \)
97 \( 1 + (-0.276 + 0.960i)T \)
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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

−22.18509023015728406896603931774, −21.76032041391707643413399138192, −20.34295177105009653750172219215, −19.63009771776149353703843937409, −19.05187677047572645627216468506, −18.21074361238373069554549650840, −17.238227260981691456080764015557, −16.74792764977285255458111835085, −15.68082751565735665485493462355, −15.01404954574459508487743871911, −13.95881224392045122691052315576, −13.36231384871107174253741240427, −12.78439156032994340728078012525, −11.71294092599866579569098056071, −10.00002317640371849571622314863, −9.34844136674561255130197193394, −8.85328451363868767745588468029, −7.93344610308856004800360340049, −6.9960913217038435048562364639, −6.1328108780820380802596172504, −5.45024596292663480221596354922, −4.151510193238708804744899917950, −2.915770550150368634863489406309, −1.676721344938078466173897259146, −0.44196393114870360770734903121, 1.85971965319361853275952105416, 2.49114080955197773880373212399, 3.54085869953366388253080496098, 4.13143137693380843692033600465, 5.37328915687436890087029809221, 6.88060365854318383645945873354, 7.5897207367145490328320320908, 8.90504123307876454950045521319, 9.58419346197592481615913503594, 10.230556314464910478933371321357, 10.62165607050123142054113046016, 12.036597233264774041908777717564, 12.77787826252394181716969636080, 13.832202139972322984879081562587, 14.35139015900096903727401477583, 15.18211927780373392639319587864, 16.527022323154349490915854486995, 17.02822350521704711411060176297, 18.13048242360859479229749958813, 18.95264779700663045132252704409, 19.63695032131184080896994404607, 20.24273510023042220964136152542, 21.0890399430412009716714017699, 21.87414149438071806213269595967, 22.549879367722991017748932585062

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