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Properties

Label 1-731-731.13-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.664 + 0.746i$
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  + (0.900 + 0.433i)2-s + (0.997 + 0.0747i)3-s + (0.623 + 0.781i)4-s + (−0.294 + 0.955i)5-s + (0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s + (0.222 + 0.974i)8-s + (0.988 + 0.149i)9-s + (−0.680 + 0.733i)10-s + (−0.781 − 0.623i)11-s + (0.563 + 0.826i)12-s + (−0.733 + 0.680i)13-s + (−0.997 + 0.0747i)14-s + (−0.365 + 0.930i)15-s + (−0.222 + 0.974i)16-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.997 + 0.0747i)3-s + (0.623 + 0.781i)4-s + (−0.294 + 0.955i)5-s + (0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s + (0.222 + 0.974i)8-s + (0.988 + 0.149i)9-s + (−0.680 + 0.733i)10-s + (−0.781 − 0.623i)11-s + (0.563 + 0.826i)12-s + (−0.733 + 0.680i)13-s + (−0.997 + 0.0747i)14-s + (−0.365 + 0.930i)15-s + (−0.222 + 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.664 + 0.746i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.664 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141200190 + 2.544083714i\)
\(L(\frac12)\) \(\approx\) \(1.141200190 + 2.544083714i\)
\(L(1)\) \(\approx\) \(1.612338015 + 1.209298082i\)
\(L(1)\) \(\approx\) \(1.612338015 + 1.209298082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
3 \( 1 + (0.997 + 0.0747i)T \)
5 \( 1 + (-0.294 + 0.955i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.781 - 0.623i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (0.988 - 0.149i)T \)
23 \( 1 + (0.930 - 0.365i)T \)
29 \( 1 + (-0.997 + 0.0747i)T \)
31 \( 1 + (0.563 + 0.826i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.563 - 0.826i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (0.930 + 0.365i)T \)
73 \( 1 + (-0.680 - 0.733i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (0.781 + 0.623i)T \)
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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

−22.323697592765465119090563543507, −21.09397132922492352011067616304, −20.55299980231567061505403759890, −20.050135813769503428179332001696, −19.37164743287694556413561337196, −18.60057347276740393691580815229, −17.193786394006327825598087250218, −16.155504139090005655554056258157, −15.45141785788489425991675084870, −14.91190863340058221458117465906, −13.602580854294797456016410862554, −13.24126642951374324358429317627, −12.57302172434227924542081580625, −11.79487788100918804059353554544, −10.28762122882656485636052116019, −9.852224705227093036427593329585, −8.90311285372696889237988432586, −7.51970993468049311735908375573, −7.1944582853813427055778222038, −5.60310093483428131957720583743, −4.81411021332371945272465752359, −3.833938993464293045966903930716, −3.08712767123178060459762289709, −2.090210586860871394667639296051, −0.825832358766966568354528210256, 2.175229773492160367257931007849, 3.01867965381706791281526460496, 3.37872992549839482431069600064, 4.623692833251470448877206089511, 5.7078342792394878686988664435, 6.86422939364670146618687990171, 7.29363989881241496149763610400, 8.3140325795090839325721449469, 9.2915332765001933567557395525, 10.32271121027178301669974474642, 11.32728280244681762768193677089, 12.32452707892675089265934413477, 13.13501899081836079310527652986, 13.94668027014317603292511826346, 14.536661901967223438715331494526, 15.455406360654880327386488865251, 15.828681334443701997801699318816, 16.75087437197974523600555558002, 18.16907121850868791759041460055, 18.9622180427948039236993047097, 19.54037655130855449268928754660, 20.54190697504304895603996061920, 21.45501520180177540863969413640, 21.982919731479618876776621896796, 22.71399764525202446265719046316

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