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Properties

Label 1-709-709.20-r0-0-0
Degree $1$
Conductor $709$
Sign $0.977 - 0.208i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (−0.964 + 0.263i)2-s + (−0.237 − 0.971i)3-s + (0.861 − 0.507i)4-s + (−0.617 + 0.786i)5-s + (0.484 + 0.874i)6-s + (0.288 − 0.957i)7-s + (−0.697 + 0.716i)8-s + (−0.887 + 0.461i)9-s + (0.388 − 0.921i)10-s + (−0.769 − 0.638i)11-s + (−0.697 − 0.716i)12-s + (0.288 + 0.957i)13-s + (−0.0266 + 0.999i)14-s + (0.910 + 0.413i)15-s + (0.484 − 0.874i)16-s + (−0.769 + 0.638i)17-s + ⋯
L(s)  = 1  + (−0.964 + 0.263i)2-s + (−0.237 − 0.971i)3-s + (0.861 − 0.507i)4-s + (−0.617 + 0.786i)5-s + (0.484 + 0.874i)6-s + (0.288 − 0.957i)7-s + (−0.697 + 0.716i)8-s + (−0.887 + 0.461i)9-s + (0.388 − 0.921i)10-s + (−0.769 − 0.638i)11-s + (−0.697 − 0.716i)12-s + (0.288 + 0.957i)13-s + (−0.0266 + 0.999i)14-s + (0.910 + 0.413i)15-s + (0.484 − 0.874i)16-s + (−0.769 + 0.638i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.977 - 0.208i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.977 - 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5849255166 - 0.06174993230i\)
\(L(\frac12)\) \(\approx\) \(0.5849255166 - 0.06174993230i\)
\(L(1)\) \(\approx\) \(0.5559335575 - 0.06651819187i\)
\(L(1)\) \(\approx\) \(0.5559335575 - 0.06651819187i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.964 + 0.263i)T \)
3 \( 1 + (-0.237 - 0.971i)T \)
5 \( 1 + (-0.617 + 0.786i)T \)
7 \( 1 + (0.288 - 0.957i)T \)
11 \( 1 + (-0.769 - 0.638i)T \)
13 \( 1 + (0.288 + 0.957i)T \)
17 \( 1 + (-0.769 + 0.638i)T \)
19 \( 1 + (-0.697 + 0.716i)T \)
23 \( 1 + (0.574 - 0.818i)T \)
29 \( 1 + (0.734 + 0.678i)T \)
31 \( 1 + (0.910 + 0.413i)T \)
37 \( 1 + (0.288 - 0.957i)T \)
41 \( 1 + (0.0797 + 0.996i)T \)
43 \( 1 + (-0.437 - 0.899i)T \)
47 \( 1 + (0.574 - 0.818i)T \)
53 \( 1 + (0.910 + 0.413i)T \)
59 \( 1 + (-0.697 + 0.716i)T \)
61 \( 1 + (0.185 + 0.982i)T \)
67 \( 1 + (0.802 + 0.596i)T \)
71 \( 1 + (0.388 + 0.921i)T \)
73 \( 1 + (-0.339 - 0.940i)T \)
79 \( 1 + (0.977 + 0.211i)T \)
83 \( 1 + (-0.833 + 0.552i)T \)
89 \( 1 + (-0.530 + 0.847i)T \)
97 \( 1 + (-0.132 - 0.991i)T \)
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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

−22.518543465757235160710381113894, −21.457895395094488960240283872098, −20.922362066513549839777490629869, −20.25535601652267564505633464075, −19.54280561092415482209387898147, −18.49646048489171678712185174369, −17.54599954411536547276588386943, −17.164725790882511501543874760893, −15.73881308583938212822654709181, −15.66434550031376211353103623688, −15.095149811864387411960368366900, −13.238680328191221356104727764322, −12.366943614409882719312669044348, −11.54940196317345947253104866090, −10.96312196088915633931031764848, −9.91612043848265554496430680420, −9.16620328884574569996075964077, −8.43730940295361130981438567327, −7.776813033790246974462610089436, −6.3670073963152264826975546675, −5.21422880529925162950023902279, −4.518607615942024009766221425373, −3.168840478347993680342000101571, −2.315194652106394207675432069000, −0.630687167265248393979498589605, 0.72835952503700647095125836695, 1.936414546706626902278644649089, 2.92940209016625589539948215097, 4.32270622829845781960481697862, 5.84398321657760812704995794548, 6.7728192917728675579399439495, 7.110812839999199241565415114695, 8.2650398824350197442057513297, 8.54625393528725874336831193116, 10.35256766373511334658084749484, 10.795573963350594347012118690522, 11.450431135737351478730192343013, 12.44531195325403482986066110511, 13.66686249996711404538591538088, 14.34671206948679339817927534203, 15.25630135435327035364483210622, 16.40907195172048942811104392405, 16.85524845755516266331247733629, 17.9559715879177577492038661499, 18.41609069260745363122409585204, 19.27278973241013234283518496546, 19.65031805760584506992693030724, 20.71009148846186039044902963137, 21.69579939975069790698987968366, 23.28173448493405359090344666552

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