Properties

Label 1-731-731.652-r0-0-0
Degree $1$
Conductor $731$
Sign $0.811 - 0.583i$
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 + (0.130 − 0.991i)3-s + i·4-s + (−0.991 − 0.130i)5-s + (−0.793 + 0.608i)6-s + (−0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.608 + 0.793i)10-s + (−0.923 − 0.382i)11-s + (0.991 + 0.130i)12-s + (−0.866 + 0.5i)13-s + (0.793 + 0.608i)14-s + (−0.258 + 0.965i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.130 − 0.991i)3-s + i·4-s + (−0.991 − 0.130i)5-s + (−0.793 + 0.608i)6-s + (−0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.608 + 0.793i)10-s + (−0.923 − 0.382i)11-s + (0.991 + 0.130i)12-s + (−0.866 + 0.5i)13-s + (0.793 + 0.608i)14-s + (−0.258 + 0.965i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.811 - 0.583i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (652, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.811 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3206743551 - 0.1033587206i\)
\(L(\frac12)\) \(\approx\) \(0.3206743551 - 0.1033587206i\)
\(L(1)\) \(\approx\) \(0.3953208654 - 0.2423396845i\)
\(L(1)\) \(\approx\) \(0.3953208654 - 0.2423396845i\)

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 \( 1 + (0.130 - 0.991i)T \)
5 \( 1 + (-0.991 - 0.130i)T \)
7 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.793 + 0.608i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.130 - 0.991i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.991 - 0.130i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.793 + 0.608i)T \)
73 \( 1 + (0.608 - 0.793i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (0.965 - 0.258i)T \)
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

−22.60847481462224264377657897850, −22.188736522718785988936380028, −20.534035338267466805893568927170, −20.18558968977595345366636452799, −19.33290386747485414579547246626, −18.636308226127085084415986081216, −17.54958939950120467724455053016, −16.68132648117875795834259242552, −15.79706789499674292557002902842, −15.702586910826241146544829996480, −14.75913283912640941854183983448, −13.92251335283198002527032201462, −12.63409494678162288145719704982, −11.595156738848771301485487472389, −10.48113890673063546491798318885, −10.049786702009155236809452366811, −9.25541656205209160051951102734, −8.12170463697585865855818907242, −7.60729777051714450626125540416, −6.532765085300587147762301553286, −5.3892072020734625991528474964, −4.63702842677587760083708276363, −3.474976014147613717639213430104, −2.512509440030712287316038797087, −0.33129796962108790824266564904, 0.65752462940379460708769216313, 2.114589936882506179053554619955, 3.00884566417207902450525885893, 3.75059621773191668142319906759, 5.24533782083225955217181178712, 6.62217506171683544419050205530, 7.50871820577025114709670606368, 7.9283087401444922117425633215, 9.02765786354422338777248321572, 9.73961247879150484947111943349, 10.98105577144956957825872037097, 11.73745264355642399530068010984, 12.39339968463152949592162098582, 13.05953813168980965697217856130, 13.8683421138881796027371384048, 15.24466574023554304623275582570, 16.20675933272782545812723226988, 16.7485151353541027915018662723, 17.9213301350843429365625895857, 18.6397965055929252862022976959, 19.14480996943100155585621729864, 19.93294899073935407092803103811, 20.278310734952991584785444596112, 21.622096153357542755037640693921, 22.40938812696431393354556728106

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