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Properties

Label 1-709-709.165-r0-0-0
Degree $1$
Conductor $709$
Sign $0.755 - 0.654i$
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  + (−0.617 − 0.786i)2-s + (0.185 − 0.982i)3-s + (−0.237 + 0.971i)4-s + (−0.769 + 0.638i)5-s + (−0.887 + 0.461i)6-s + (−0.998 + 0.0532i)7-s + (0.910 − 0.413i)8-s + (−0.931 − 0.364i)9-s + (0.977 + 0.211i)10-s + (−0.987 − 0.159i)11-s + (0.910 + 0.413i)12-s + (−0.998 − 0.0532i)13-s + (0.658 + 0.752i)14-s + (0.484 + 0.874i)15-s + (−0.887 − 0.461i)16-s + (−0.987 + 0.159i)17-s + ⋯
L(s)  = 1  + (−0.617 − 0.786i)2-s + (0.185 − 0.982i)3-s + (−0.237 + 0.971i)4-s + (−0.769 + 0.638i)5-s + (−0.887 + 0.461i)6-s + (−0.998 + 0.0532i)7-s + (0.910 − 0.413i)8-s + (−0.931 − 0.364i)9-s + (0.977 + 0.211i)10-s + (−0.987 − 0.159i)11-s + (0.910 + 0.413i)12-s + (−0.998 − 0.0532i)13-s + (0.658 + 0.752i)14-s + (0.484 + 0.874i)15-s + (−0.887 − 0.461i)16-s + (−0.987 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4080116791 - 0.1522188539i\)
\(L(\frac12)\) \(\approx\) \(0.4080116791 - 0.1522188539i\)
\(L(1)\) \(\approx\) \(0.4575445800 - 0.2377016959i\)
\(L(1)\) \(\approx\) \(0.4575445800 - 0.2377016959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.617 - 0.786i)T \)
3 \( 1 + (0.185 - 0.982i)T \)
5 \( 1 + (-0.769 + 0.638i)T \)
7 \( 1 + (-0.998 + 0.0532i)T \)
11 \( 1 + (-0.987 - 0.159i)T \)
13 \( 1 + (-0.998 - 0.0532i)T \)
17 \( 1 + (-0.987 + 0.159i)T \)
19 \( 1 + (0.910 - 0.413i)T \)
23 \( 1 + (0.734 + 0.678i)T \)
29 \( 1 + (0.288 - 0.957i)T \)
31 \( 1 + (0.484 + 0.874i)T \)
37 \( 1 + (-0.998 + 0.0532i)T \)
41 \( 1 + (-0.833 - 0.552i)T \)
43 \( 1 + (-0.339 - 0.940i)T \)
47 \( 1 + (0.734 + 0.678i)T \)
53 \( 1 + (0.484 + 0.874i)T \)
59 \( 1 + (0.910 - 0.413i)T \)
61 \( 1 + (0.949 + 0.314i)T \)
67 \( 1 + (-0.0266 + 0.999i)T \)
71 \( 1 + (0.977 - 0.211i)T \)
73 \( 1 + (0.0797 + 0.996i)T \)
79 \( 1 + (0.861 + 0.507i)T \)
83 \( 1 + (0.994 + 0.106i)T \)
89 \( 1 + (0.574 - 0.818i)T \)
97 \( 1 + (-0.437 + 0.899i)T \)
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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

−22.72117067279007512350796596073, −22.197784362839200724024247446998, −20.81700317854009295480387343749, −20.09912501608015427219662823953, −19.550399437025784256006348695747, −18.69689452547252025581123244107, −17.5858360729209132184349245773, −16.51649637441785888727002938623, −16.33135714777058351063348164499, −15.40128704864887946120556072407, −15.01719405348213666122655419723, −13.78243219557132545438422305179, −12.89919840829840053872459983413, −11.725913387097316201617442754941, −10.628709700237589123367456481138, −9.917743186369240413082192380367, −9.18055887034021350912576162298, −8.41758681788936099498245182448, −7.53057273200557879187438408584, −6.59994679209870528791299940504, −5.18341303673200762701001786285, −4.86888963750525198522268946695, −3.63357528203476931960047750903, −2.451386911815462321310402210126, −0.424261820607292156344150035134, 0.650415692202624134181685592343, 2.35478037627858708620877547321, 2.8469978435301973393820320947, 3.72571964708387497439811955194, 5.22907130399156833243830694588, 6.84023438194969736717884622513, 7.1727660610965310436981007506, 8.111001385792818724815492184653, 8.968610517267005685248748885358, 10.00932514229786235803849040511, 10.85315235488209849912234333163, 11.80999659828976603583376530238, 12.34854719729318489413166107997, 13.283081797395390642659987388669, 13.845860608737720838522042601518, 15.33009203394007331082468298272, 15.91847031279390688808544994676, 17.215277937656592691766982004778, 17.80830200440007042666645457198, 18.82486084856857223256333366834, 19.142178551103379157646450180009, 19.819496753364432594079894161187, 20.49857175998611924904775042520, 21.80879512476889342326538126806, 22.482287545638146013512813927211

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