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Properties

Label 1-731-731.371-r0-0-0
Degree $1$
Conductor $731$
Sign $0.961 - 0.275i$
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 + (−0.666 − 0.745i)3-s + (−0.781 − 0.623i)4-s + (−0.815 − 0.578i)5-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.846 + 0.532i)8-s + (−0.111 + 0.993i)9-s + (−0.815 + 0.578i)10-s + (0.875 + 0.483i)11-s + (0.0560 + 0.998i)12-s + (−0.974 − 0.222i)13-s + (−0.998 + 0.0560i)14-s + (0.111 + 0.993i)15-s + (0.222 + 0.974i)16-s + ⋯
L(s)  = 1  + (0.330 − 0.943i)2-s + (−0.666 − 0.745i)3-s + (−0.781 − 0.623i)4-s + (−0.815 − 0.578i)5-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.846 + 0.532i)8-s + (−0.111 + 0.993i)9-s + (−0.815 + 0.578i)10-s + (0.875 + 0.483i)11-s + (0.0560 + 0.998i)12-s + (−0.974 − 0.222i)13-s + (−0.998 + 0.0560i)14-s + (0.111 + 0.993i)15-s + (0.222 + 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3080392449 - 0.04326304381i\)
\(L(\frac12)\) \(\approx\) \(0.3080392449 - 0.04326304381i\)
\(L(1)\) \(\approx\) \(0.4350928891 - 0.4468057573i\)
\(L(1)\) \(\approx\) \(0.4350928891 - 0.4468057573i\)

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 \( 1 + (-0.666 - 0.745i)T \)
5 \( 1 + (-0.815 - 0.578i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.875 + 0.483i)T \)
13 \( 1 + (-0.974 - 0.222i)T \)
19 \( 1 + (0.111 + 0.993i)T \)
23 \( 1 + (-0.875 - 0.483i)T \)
29 \( 1 + (0.0560 + 0.998i)T \)
31 \( 1 + (-0.998 + 0.0560i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.745 - 0.666i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.846 - 0.532i)T \)
59 \( 1 + (-0.532 + 0.846i)T \)
61 \( 1 + (0.0560 - 0.998i)T \)
67 \( 1 + (-0.623 + 0.781i)T \)
71 \( 1 + (0.875 - 0.483i)T \)
73 \( 1 + (-0.815 - 0.578i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.943 - 0.330i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
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

−22.43229996659370692537450546280, −21.88333284118680184173464667580, −21.694114902880352280860500407366, −20.07995236791466658922846573341, −19.18158997473579113595113500670, −18.29962939165637723483426924062, −17.48977062287549027882370410793, −16.59718817737469735448672261505, −15.993578580354339649725102395418, −15.15198691711135592627626315207, −14.83595305467031328655756296588, −13.764293981879267112672566357675, −12.40129020307800193715259562929, −11.88045241992271137130920587630, −11.14228355352761653905165815479, −9.69949068186790745897452185816, −9.19658884852558702589158595916, −8.11192215536829581592004038500, −7.0106365327868070321405307519, −6.30146387043411063194655363987, −5.4899037241323894597805379549, −4.46965812085083867588020571841, −3.71527869344984218583819020340, −2.755396791444163188555541370275, −0.17876744507663239756010745570, 1.00238594935494194214819677086, 1.908669428910746225948755752135, 3.39256680746470828859041759471, 4.284187082835591231639422521024, 5.02562076838689404815996573262, 6.174588073418336435545871482018, 7.22892566778494944833262408708, 8.01830933705464226430492787887, 9.250762135063513293195297825056, 10.21401256051006652549818235079, 10.98178323389544783458408608444, 12.041009961450075441274384949004, 12.31882096226940462256921565741, 13.06024844815701321537817041330, 14.083884888960717885594878015253, 14.79257786109497960046247470208, 16.23768909807362155270153712288, 16.89331552390109465080768783031, 17.67010640693534137364206825238, 18.67593882048286661133994254552, 19.43011141255508353660541348478, 20.07714885289571194499692306656, 20.46536052183809749065061303480, 22.05199366658318481535920515933, 22.41309877390423456835995917816

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