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Properties

Label 1-731-731.159-r0-0-0
Degree $1$
Conductor $731$
Sign $0.0420 - 0.999i$
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 + (0.312 + 0.949i)3-s + (−0.781 + 0.623i)4-s + (0.0933 + 0.995i)5-s + (−0.793 + 0.608i)6-s + (−0.991 + 0.130i)7-s + (−0.846 − 0.532i)8-s + (−0.804 + 0.593i)9-s + (−0.908 + 0.416i)10-s + (−0.875 + 0.483i)11-s + (−0.836 − 0.547i)12-s + (0.680 + 0.733i)13-s + (−0.450 − 0.892i)14-s + (−0.916 + 0.399i)15-s + (0.222 − 0.974i)16-s + ⋯
L(s)  = 1  + (0.330 + 0.943i)2-s + (0.312 + 0.949i)3-s + (−0.781 + 0.623i)4-s + (0.0933 + 0.995i)5-s + (−0.793 + 0.608i)6-s + (−0.991 + 0.130i)7-s + (−0.846 − 0.532i)8-s + (−0.804 + 0.593i)9-s + (−0.908 + 0.416i)10-s + (−0.875 + 0.483i)11-s + (−0.836 − 0.547i)12-s + (0.680 + 0.733i)13-s + (−0.450 − 0.892i)14-s + (−0.916 + 0.399i)15-s + (0.222 − 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0420 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0420 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0420 - 0.999i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.0420 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7579025685 + 0.7266822838i\)
\(L(\frac12)\) \(\approx\) \(-0.7579025685 + 0.7266822838i\)
\(L(1)\) \(\approx\) \(0.3316719954 + 0.9464855794i\)
\(L(1)\) \(\approx\) \(0.3316719954 + 0.9464855794i\)

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 \( 1 + (0.312 + 0.949i)T \)
5 \( 1 + (0.0933 + 0.995i)T \)
7 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (-0.875 + 0.483i)T \)
13 \( 1 + (0.680 + 0.733i)T \)
19 \( 1 + (0.804 + 0.593i)T \)
23 \( 1 + (-0.0186 + 0.999i)T \)
29 \( 1 + (0.892 - 0.450i)T \)
31 \( 1 + (-0.547 + 0.836i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.745 - 0.666i)T \)
47 \( 1 + (0.781 - 0.623i)T \)
53 \( 1 + (-0.0373 - 0.999i)T \)
59 \( 1 + (-0.532 - 0.846i)T \)
61 \( 1 + (-0.836 + 0.547i)T \)
67 \( 1 + (0.988 - 0.149i)T \)
71 \( 1 + (0.0186 + 0.999i)T \)
73 \( 1 + (-0.908 - 0.416i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (-0.757 + 0.652i)T \)
89 \( 1 + (-0.997 - 0.0747i)T \)
97 \( 1 + (0.960 + 0.276i)T \)
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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

−21.864482165102218171337854853891, −20.79747221405731969442027760943, −20.21525828642068414874045483191, −19.73836457934435113683193312509, −18.70312604228148175309902010208, −18.24624758605043339348592929575, −17.24678124684086725461567090571, −16.16775291097437994442570167408, −15.300959727975030909658644775564, −13.935215978438448121337499454844, −13.4129735084551235535867613875, −12.81627545771358833567847350381, −12.30291263736165214526060273882, −11.23376351388648870094886616311, −10.25290687452863433769193983024, −9.24737837747046613467870705618, −8.580970021140175207147264796161, −7.6849758439202874053127992044, −6.217509571178201115234401768203, −5.631163316152340490916327660556, −4.47332979493962166755471544827, −3.18057659599337406730744034574, −2.66232315595385335669939914884, −1.240391103078360391627894303408, −0.45894958126400653215225741534, 2.46270195907047051649958656689, 3.40985179140709085678546861278, 3.973477671797404877128062692851, 5.29313200582188238823876655347, 5.978883968316991246719634632303, 6.99125813028573944411284011285, 7.77871813440435640304109700301, 8.9179131152939775290026319893, 9.69126268933666926826576212180, 10.34909163663787845735424872303, 11.47674370211254095089287152843, 12.62469955883456515109221960880, 13.73790410769064254198905185646, 14.115455221658832086781712535583, 15.13025260136263341159949087273, 15.89169383014988544439760091875, 16.08367738303628332728177158465, 17.29961637852699680193148270057, 18.19099736509511083547405665119, 18.93224787662330098209647474298, 19.89917300823574771206305022503, 21.193244691695495434914415954971, 21.54275503906554176175055979813, 22.526281139130345892642673811646, 22.98638769379200347701021885610

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