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Properties

Label 1-712-712.253-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.740 + 0.671i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.800 + 0.599i)3-s + (−0.540 − 0.841i)5-s + (0.212 − 0.977i)7-s + (0.281 + 0.959i)9-s + (0.841 + 0.540i)11-s + (−0.599 + 0.800i)13-s + (0.0713 − 0.997i)15-s + (−0.755 − 0.654i)17-s + (−0.877 − 0.479i)19-s + (0.755 − 0.654i)21-s + (−0.479 + 0.877i)23-s + (−0.415 + 0.909i)25-s + (−0.349 + 0.936i)27-s + (0.977 + 0.212i)29-s + (−0.479 − 0.877i)31-s + ⋯
L(s)  = 1  + (0.800 + 0.599i)3-s + (−0.540 − 0.841i)5-s + (0.212 − 0.977i)7-s + (0.281 + 0.959i)9-s + (0.841 + 0.540i)11-s + (−0.599 + 0.800i)13-s + (0.0713 − 0.997i)15-s + (−0.755 − 0.654i)17-s + (−0.877 − 0.479i)19-s + (0.755 − 0.654i)21-s + (−0.479 + 0.877i)23-s + (−0.415 + 0.909i)25-s + (−0.349 + 0.936i)27-s + (0.977 + 0.212i)29-s + (−0.479 − 0.877i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3343130700 + 0.8660995572i\)
\(L(\frac12)\) \(\approx\) \(0.3343130700 + 0.8660995572i\)
\(L(1)\) \(\approx\) \(1.075534279 + 0.1285225293i\)
\(L(1)\) \(\approx\) \(1.075534279 + 0.1285225293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.800 + 0.599i)T \)
5 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (0.212 - 0.977i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.599 + 0.800i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (-0.877 - 0.479i)T \)
23 \( 1 + (-0.479 + 0.877i)T \)
29 \( 1 + (0.977 + 0.212i)T \)
31 \( 1 + (-0.479 - 0.877i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.599 - 0.800i)T \)
43 \( 1 + (0.977 - 0.212i)T \)
47 \( 1 + (-0.989 - 0.142i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (-0.800 + 0.599i)T \)
61 \( 1 + (0.936 + 0.349i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.540 + 0.841i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (-0.281 + 0.959i)T \)
83 \( 1 + (0.0713 + 0.997i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−21.97963280797644546437336327886, −21.44687199236431853170698056380, −20.17819907851284141807607016901, −19.4887443009119792488625592635, −19.023183335325079484712839004875, −18.1156799753124671740759665100, −17.54473900327519755188241352557, −16.139183939446516599979094385436, −15.20205450521630566996044759837, −14.66182749177142789267888142785, −14.12592099189540470427143175549, −12.76917449873918357113919536496, −12.289282822528052235429297587920, −11.32024072504470665653537596804, −10.37591668470842122819800660006, −9.223680762804328664346578911787, −8.33883586474277023128785714547, −7.872317771930630448979371858089, −6.5308425463828883944264897633, −6.198323704686270101622947498586, −4.530688465840260826259578332957, −3.457971088081519654554273813717, −2.65711796613964572683891631275, −1.77713532393895443909318903414, −0.182180899316484136500796669572, 1.29778561416159586900439980046, 2.37364593555312116911462904109, 3.87529020574273626473637025731, 4.30809296236030002134034440117, 5.005307530759121709195173425292, 6.75892820670947186798591806931, 7.50105385733922357164742544364, 8.42701640851223337643037042246, 9.275093317701889135138861677, 9.84946747928103519830465949478, 11.0624807132546703384710511519, 11.79597419470075282607424390387, 12.933885246960038847051740128892, 13.72507251018368767862243485267, 14.470400440878275847907359377512, 15.33171298964999988650925151607, 16.1279797958847816629751371986, 16.92347324808074538106305591130, 17.50888445154461559230102075798, 19.03417756640759275008655488800, 19.80371729500231180797301982046, 20.09738153512038982534111449170, 20.92274898617052593479927634623, 21.77340387957919485586570294208, 22.615319045383703219539193731

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