↑

Properties

Label 1-731-731.9-r0-0-0
Degree $1$
Conductor $731$
Sign $0.420 + 0.907i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.974 + 0.222i)2-s + (0.467 + 0.884i)3-s + (0.900 + 0.433i)4-s + (0.399 − 0.916i)5-s + (0.258 + 0.965i)6-s + (0.965 + 0.258i)7-s + (0.781 + 0.623i)8-s + (−0.563 + 0.826i)9-s + (0.593 − 0.804i)10-s + (−0.330 + 0.943i)11-s + (0.0373 + 0.999i)12-s + (0.988 + 0.149i)13-s + (0.884 + 0.467i)14-s + (0.997 − 0.0747i)15-s + (0.623 + 0.781i)16-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.467 + 0.884i)3-s + (0.900 + 0.433i)4-s + (0.399 − 0.916i)5-s + (0.258 + 0.965i)6-s + (0.965 + 0.258i)7-s + (0.781 + 0.623i)8-s + (−0.563 + 0.826i)9-s + (0.593 − 0.804i)10-s + (−0.330 + 0.943i)11-s + (0.0373 + 0.999i)12-s + (0.988 + 0.149i)13-s + (0.884 + 0.467i)14-s + (0.997 − 0.0747i)15-s + (0.623 + 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.420 + 0.907i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.420 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.112092652 + 1.988002373i\)
\(L(\frac12)\) \(\approx\) \(3.112092652 + 1.988002373i\)
\(L(1)\) \(\approx\) \(2.287628122 + 0.9155677219i\)
\(L(1)\) \(\approx\) \(2.287628122 + 0.9155677219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.974 + 0.222i)T \)
3 \( 1 + (0.467 + 0.884i)T \)
5 \( 1 + (0.399 - 0.916i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (-0.330 + 0.943i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (-0.563 - 0.826i)T \)
23 \( 1 + (-0.652 - 0.757i)T \)
29 \( 1 + (-0.884 - 0.467i)T \)
31 \( 1 + (-0.0373 - 0.999i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.532 - 0.846i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.149 + 0.988i)T \)
59 \( 1 + (-0.781 + 0.623i)T \)
61 \( 1 + (0.0373 - 0.999i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (-0.652 + 0.757i)T \)
73 \( 1 + (0.593 + 0.804i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + (0.294 - 0.955i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
97 \( 1 + (0.943 + 0.330i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.43162781440887560682031854578, −21.400473456788529578640972379679, −20.96969370567819744323674814184, −20.07448480307066270787015998931, −19.12227155558675374672950801012, −18.47574340350186228705330170864, −17.77645044552821255841399301614, −16.6130531386294703505131890075, −15.433496504705534285003145349512, −14.673740420771497989933764339415, −13.90344652749089192060170945925, −13.6493064281884125380045585900, −12.628218045578307331222260333927, −11.559386891763880743219113085411, −10.99762693817489335336761106114, −10.188905897545215491302744762473, −8.648560025104354468145799302321, −7.795464279498729806066736976832, −6.94699520142934798354198431532, −6.03667928636773823863279954991, −5.42867661310321728903737872858, −3.7761572552162717237256659387, −3.23909021620478946200138377207, −2.035653108808464159849961215546, −1.397800832174481594168830817674, 1.80234170090722854258032116282, 2.43856573294871034157214651959, 3.96826330169522854542607636480, 4.503869729390441699830607001462, 5.251744307788627256523814826, 6.06188010838323583842569628365, 7.52022121780436995439348808614, 8.37693049222419224495021229087, 9.05925267264440215115745591554, 10.2781678604666627592773657189, 11.0913708305350770997650094993, 12.002115589939387692016189767907, 12.96555127232711695328184537388, 13.7287993421643369414841555451, 14.43614794828319702146218936351, 15.47131053701453782211693656645, 15.66778833308253718298075014508, 16.92842218072307716328519916715, 17.32454571560150966121706828936, 18.64681473548761509250934025645, 20.13394283744312394457401141420, 20.48356805019373616229293548459, 21.03999727202281419734960135707, 21.71010068713644722554743087135, 22.53188392274402962237145558814

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