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Properties

Label 1-731-731.609-r0-0-0
Degree $1$
Conductor $731$
Sign $0.206 - 0.978i$
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  + (−0.707 + 0.707i)2-s + (−0.793 + 0.608i)3-s − i·4-s + (−0.608 − 0.793i)5-s + (0.130 − 0.991i)6-s + (−0.608 + 0.793i)7-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (0.991 + 0.130i)10-s + (0.923 − 0.382i)11-s + (0.608 + 0.793i)12-s + (0.866 − 0.5i)13-s + (−0.130 − 0.991i)14-s + (0.965 + 0.258i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.793 + 0.608i)3-s − i·4-s + (−0.608 − 0.793i)5-s + (0.130 − 0.991i)6-s + (−0.608 + 0.793i)7-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (0.991 + 0.130i)10-s + (0.923 − 0.382i)11-s + (0.608 + 0.793i)12-s + (0.866 − 0.5i)13-s + (−0.130 − 0.991i)14-s + (0.965 + 0.258i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2369157408 - 0.1920698160i\)
\(L(\frac12)\) \(\approx\) \(0.2369157408 - 0.1920698160i\)
\(L(1)\) \(\approx\) \(0.4600260082 + 0.09269809801i\)
\(L(1)\) \(\approx\) \(0.4600260082 + 0.09269809801i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.793 + 0.608i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 + (-0.608 + 0.793i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (0.130 - 0.991i)T \)
29 \( 1 + (-0.991 + 0.130i)T \)
31 \( 1 + (-0.793 + 0.608i)T \)
37 \( 1 + (-0.793 + 0.608i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.608 - 0.793i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.130 - 0.991i)T \)
73 \( 1 + (0.991 - 0.130i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.258 - 0.965i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−22.72943746600115670979426814234, −22.070185129975051356026946600896, −20.986394439615709606754594734987, −19.9467023274486507938768325432, −19.30701913422854259361232309293, −18.74934830255935677515009517797, −17.99607290945173976149255118543, −17.07629032157144857473748705438, −16.536668703561139997175169793567, −15.67721097633154232469793060738, −14.26754528212215690449147792261, −13.38695606428785368777169696454, −12.53816420036105209153111288466, −11.67910069261569157400360065085, −11.13153757463972097499131938520, −10.38267395545641570278815490567, −9.51284381309533112872911377363, −8.29701383010404016220704538143, −7.17046101860944131464614252218, −7.006060343632466076941877474578, −5.83677870494200964896985893799, −3.87120642994636618793336921504, −3.82694851625420119150243682567, −2.13992992336396836497677785164, −1.17917391485789825913982332868, 0.247120852960029511422865703813, 1.38787284882881810802338749046, 3.31092923695254182865352677177, 4.44728293641979062918226146404, 5.32633667276016430586309329335, 6.140036196174661919061788966917, 6.83282474825127780764881778489, 8.20705712647675321512271477494, 9.06052524652707670004133891468, 9.36263275181237190282242928965, 10.71783115008034617359885141197, 11.31126610648398719105448594894, 12.290948734018881005313319127057, 13.1507939996764346184471266305, 14.56586798123577254571764884224, 15.436880059939021846979010298514, 15.889983363670129943386363675160, 16.617525988017377305879652578942, 17.18792722893727122555451724824, 18.207544816923149292038783562373, 18.92224080300449721620355180345, 19.860908905016220027658750922932, 20.550695310821113618862284542837, 21.70457435710439682930927914706, 22.55257228782772852387267225416

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