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Properties

Label 1-709-709.336-r0-0-0
Degree $1$
Conductor $709$
Sign $0.454 - 0.890i$
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  + (−0.437 + 0.899i)2-s + (−0.769 + 0.638i)3-s + (−0.617 − 0.786i)4-s + (−0.339 − 0.940i)5-s + (−0.237 − 0.971i)6-s + (−0.0266 − 0.999i)7-s + (0.977 − 0.211i)8-s + (0.185 − 0.982i)9-s + (0.994 + 0.106i)10-s + (0.0797 − 0.996i)11-s + (0.977 + 0.211i)12-s + (−0.0266 + 0.999i)13-s + (0.910 + 0.413i)14-s + (0.861 + 0.507i)15-s + (−0.237 + 0.971i)16-s + (0.0797 + 0.996i)17-s + ⋯
L(s)  = 1  + (−0.437 + 0.899i)2-s + (−0.769 + 0.638i)3-s + (−0.617 − 0.786i)4-s + (−0.339 − 0.940i)5-s + (−0.237 − 0.971i)6-s + (−0.0266 − 0.999i)7-s + (0.977 − 0.211i)8-s + (0.185 − 0.982i)9-s + (0.994 + 0.106i)10-s + (0.0797 − 0.996i)11-s + (0.977 + 0.211i)12-s + (−0.0266 + 0.999i)13-s + (0.910 + 0.413i)14-s + (0.861 + 0.507i)15-s + (−0.237 + 0.971i)16-s + (0.0797 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (336, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.454 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4654302897 - 0.2851467202i\)
\(L(\frac12)\) \(\approx\) \(0.4654302897 - 0.2851467202i\)
\(L(1)\) \(\approx\) \(0.5799996477 + 0.06523681153i\)
\(L(1)\) \(\approx\) \(0.5799996477 + 0.06523681153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.437 + 0.899i)T \)
3 \( 1 + (-0.769 + 0.638i)T \)
5 \( 1 + (-0.339 - 0.940i)T \)
7 \( 1 + (-0.0266 - 0.999i)T \)
11 \( 1 + (0.0797 - 0.996i)T \)
13 \( 1 + (-0.0266 + 0.999i)T \)
17 \( 1 + (0.0797 + 0.996i)T \)
19 \( 1 + (0.977 - 0.211i)T \)
23 \( 1 + (-0.931 - 0.364i)T \)
29 \( 1 + (0.802 - 0.596i)T \)
31 \( 1 + (0.861 + 0.507i)T \)
37 \( 1 + (-0.0266 - 0.999i)T \)
41 \( 1 + (0.288 - 0.957i)T \)
43 \( 1 + (0.574 - 0.818i)T \)
47 \( 1 + (-0.931 - 0.364i)T \)
53 \( 1 + (0.861 + 0.507i)T \)
59 \( 1 + (0.977 - 0.211i)T \)
61 \( 1 + (-0.987 - 0.159i)T \)
67 \( 1 + (-0.697 - 0.716i)T \)
71 \( 1 + (0.994 - 0.106i)T \)
73 \( 1 + (0.734 + 0.678i)T \)
79 \( 1 + (-0.964 - 0.263i)T \)
83 \( 1 + (-0.998 - 0.0532i)T \)
89 \( 1 + (-0.887 + 0.461i)T \)
97 \( 1 + (-0.530 - 0.847i)T \)
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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

−22.63906684999546461007148718828, −22.18067358459185825213353397752, −21.195887669875058878627416032006, −20.05099083792742444545316809612, −19.45312281890161921163599925945, −18.432219797965668038562995280307, −18.06643905586546452277910574166, −17.64746249763282492740746325821, −16.2824758450165665767623069850, −15.51915234911066725381454957019, −14.3588618032791718041765663244, −13.37623606093370453381055012713, −12.41724593843201153333246470583, −11.821399793227602352360256790, −11.341075732502206637099627001587, −10.15323061087695108429848427461, −9.715496324617117397914328469584, −8.17348911658058943606639250334, −7.61241654484269372846497106763, −6.64244537993633825655756216080, −5.48954986645520922721744740817, −4.54353013860886528824159155746, −3.03435570373846539361516703067, −2.464219071768645743607076143390, −1.23005589884892661493291471494, 0.41385583855777330391806230146, 1.33377546939031029226244841026, 3.85295113042271498694425662935, 4.29079986362936445691098348543, 5.31893078989607759304690421290, 6.13235594313612758159994394100, 7.02959555919180301048791502140, 8.12733263809197800829652926605, 8.89552466913224071776340382890, 9.79219644942560681756928136431, 10.57191601233420749386649158447, 11.47796761303834049091626853664, 12.447108354957730916626826654, 13.69530376006470078200473304072, 14.20298654011222164083622200304, 15.55757319841204661944431951421, 16.092321523062844654566216210870, 16.70962071320649618752820561624, 17.204327223372470706469829650449, 18.05742872289892239661321544575, 19.26773431100049170882054170487, 19.8415704220915108304195320236, 20.9598082131234652073411007471, 21.697104197379616881755394488799, 22.77139459388270825025205940148

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