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Properties

Label 1-709-709.44-r0-0-0
Degree $1$
Conductor $709$
Sign $0.839 - 0.543i$
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  + (0.949 − 0.314i)2-s + (−0.0266 + 0.999i)3-s + (0.802 − 0.596i)4-s + (−0.697 − 0.716i)5-s + (0.288 + 0.957i)6-s + (−0.617 + 0.786i)7-s + (0.574 − 0.818i)8-s + (−0.998 − 0.0532i)9-s + (−0.887 − 0.461i)10-s + (0.910 − 0.413i)11-s + (0.574 + 0.818i)12-s + (−0.617 − 0.786i)13-s + (−0.339 + 0.940i)14-s + (0.734 − 0.678i)15-s + (0.288 − 0.957i)16-s + (0.910 + 0.413i)17-s + ⋯
L(s)  = 1  + (0.949 − 0.314i)2-s + (−0.0266 + 0.999i)3-s + (0.802 − 0.596i)4-s + (−0.697 − 0.716i)5-s + (0.288 + 0.957i)6-s + (−0.617 + 0.786i)7-s + (0.574 − 0.818i)8-s + (−0.998 − 0.0532i)9-s + (−0.887 − 0.461i)10-s + (0.910 − 0.413i)11-s + (0.574 + 0.818i)12-s + (−0.617 − 0.786i)13-s + (−0.339 + 0.940i)14-s + (0.734 − 0.678i)15-s + (0.288 − 0.957i)16-s + (0.910 + 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.074641790 - 0.6128511360i\)
\(L(\frac12)\) \(\approx\) \(2.074641790 - 0.6128511360i\)
\(L(1)\) \(\approx\) \(1.605381667 - 0.1639329756i\)
\(L(1)\) \(\approx\) \(1.605381667 - 0.1639329756i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.949 - 0.314i)T \)
3 \( 1 + (-0.0266 + 0.999i)T \)
5 \( 1 + (-0.697 - 0.716i)T \)
7 \( 1 + (-0.617 + 0.786i)T \)
11 \( 1 + (0.910 - 0.413i)T \)
13 \( 1 + (-0.617 - 0.786i)T \)
17 \( 1 + (0.910 + 0.413i)T \)
19 \( 1 + (0.574 - 0.818i)T \)
23 \( 1 + (0.994 + 0.106i)T \)
29 \( 1 + (-0.964 - 0.263i)T \)
31 \( 1 + (0.734 - 0.678i)T \)
37 \( 1 + (-0.617 + 0.786i)T \)
41 \( 1 + (0.861 + 0.507i)T \)
43 \( 1 + (0.388 - 0.921i)T \)
47 \( 1 + (0.994 + 0.106i)T \)
53 \( 1 + (0.734 - 0.678i)T \)
59 \( 1 + (0.574 - 0.818i)T \)
61 \( 1 + (0.658 - 0.752i)T \)
67 \( 1 + (-0.437 + 0.899i)T \)
71 \( 1 + (-0.887 + 0.461i)T \)
73 \( 1 + (0.977 + 0.211i)T \)
79 \( 1 + (-0.931 + 0.364i)T \)
83 \( 1 + (-0.237 + 0.971i)T \)
89 \( 1 + (-0.833 + 0.552i)T \)
97 \( 1 + (-0.987 + 0.159i)T \)
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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

−22.77019634447417781825885612662, −22.45354047011146460964331939337, −21.10033765198080974055419071566, −20.13540603160800702344416880151, −19.43893066667854042300371794507, −18.908478210781385770902229928084, −17.65803342851539622754552348260, −16.7680817662404955856625238083, −16.247473436264055747941890629071, −14.92852555733705771186407213795, −14.281196506007497650334580600449, −13.83337585034018282140948095889, −12.6235199528165000353942552700, −12.09495141672822384014496378386, −11.40569294068587906991060215825, −10.38844894027283545221300123098, −9.03475240543914429085614504562, −7.54304027221664531807160978246, −7.29919020982879955263062570155, −6.63132202530064801505312109644, −5.65676014123393250781791485982, −4.30451075205298911722783521559, −3.47516420843580865311945486956, −2.63311608792695711252515934010, −1.28349879778953900334049280805, 0.859093689641904115696438987617, 2.659474405279113421320169903898, 3.42984652520878146023903270795, 4.18232514373371044067839735594, 5.29651758873928073818828368772, 5.661443907761670157946175190276, 6.98410309631077732052150293098, 8.2693827458775487294223437455, 9.29121271808797383208252059051, 9.90988423857933954197389952002, 11.13453721645361429471876030922, 11.76001092508892309212889060137, 12.46373009051789851983409390388, 13.32870836520070987367625709409, 14.49836681632203867918327422723, 15.2490129007138374914818520126, 15.65988130709151530047332982214, 16.5986791455894620032504199423, 17.19427294525926864382770610342, 19.06256939588185765616352272104, 19.43224586777019843404764932031, 20.344212779450799340782067883399, 20.93398743643493608130729764578, 21.90440869669172408662875021575, 22.39354275828783756414995714089

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