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Properties

Label 1-709-709.88-r1-0-0
Degree $1$
Conductor $709$
Sign $0.165 - 0.986i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (0.952 − 0.305i)2-s + (−0.999 + 0.0177i)3-s + (0.813 − 0.582i)4-s + (0.870 + 0.492i)5-s + (−0.946 + 0.322i)6-s + (−0.903 + 0.429i)7-s + (0.596 − 0.802i)8-s + (0.999 − 0.0354i)9-s + (0.979 + 0.202i)10-s + (0.722 − 0.691i)11-s + (−0.802 + 0.596i)12-s + (−0.996 + 0.0797i)13-s + (−0.728 + 0.684i)14-s + (−0.879 − 0.476i)15-s + (0.322 − 0.946i)16-s + (0.280 + 0.959i)17-s + ⋯
L(s)  = 1  + (0.952 − 0.305i)2-s + (−0.999 + 0.0177i)3-s + (0.813 − 0.582i)4-s + (0.870 + 0.492i)5-s + (−0.946 + 0.322i)6-s + (−0.903 + 0.429i)7-s + (0.596 − 0.802i)8-s + (0.999 − 0.0354i)9-s + (0.979 + 0.202i)10-s + (0.722 − 0.691i)11-s + (−0.802 + 0.596i)12-s + (−0.996 + 0.0797i)13-s + (−0.728 + 0.684i)14-s + (−0.879 − 0.476i)15-s + (0.322 − 0.946i)16-s + (0.280 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.165 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.143980534 - 1.813344882i\)
\(L(\frac12)\) \(\approx\) \(2.143980534 - 1.813344882i\)
\(L(1)\) \(\approx\) \(1.488574016 - 0.3931319566i\)
\(L(1)\) \(\approx\) \(1.488574016 - 0.3931319566i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.952 - 0.305i)T \)
3 \( 1 + (-0.999 + 0.0177i)T \)
5 \( 1 + (0.870 + 0.492i)T \)
7 \( 1 + (-0.903 + 0.429i)T \)
11 \( 1 + (0.722 - 0.691i)T \)
13 \( 1 + (-0.996 + 0.0797i)T \)
17 \( 1 + (0.280 + 0.959i)T \)
19 \( 1 + (0.115 - 0.993i)T \)
23 \( 1 + (-0.0709 - 0.997i)T \)
29 \( 1 + (-0.645 - 0.764i)T \)
31 \( 1 + (0.522 + 0.852i)T \)
37 \( 1 + (0.429 + 0.903i)T \)
41 \( 1 + (-0.347 - 0.937i)T \)
43 \( 1 + (-0.710 - 0.703i)T \)
47 \( 1 + (0.437 - 0.899i)T \)
53 \( 1 + (0.999 - 0.0266i)T \)
59 \( 1 + (0.802 + 0.596i)T \)
61 \( 1 + (-0.537 + 0.842i)T \)
67 \( 1 + (-0.954 + 0.297i)T \)
71 \( 1 + (0.979 - 0.202i)T \)
73 \( 1 + (0.786 - 0.617i)T \)
79 \( 1 + (0.962 - 0.271i)T \)
83 \( 1 + (-0.159 - 0.987i)T \)
89 \( 1 + (-0.610 - 0.792i)T \)
97 \( 1 + (-0.807 - 0.589i)T \)
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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

−22.59056987598135344648756266894, −22.054778793965462505205715353394, −21.24074169624045919759031390670, −20.35714025889090600173804487089, −19.597129037859231828189243762139, −18.24211507189036436146042174565, −17.323461204990909920455635272, −16.71328808004143744178718992362, −16.30784123770710365963128750616, −15.196065930592525469874463032203, −14.21482275419373030124150878737, −13.38222754575238089351077353280, −12.57508163078148066471102665034, −12.13867270083046046405449147078, −11.124397028161954620109143231504, −9.82260647231600497568210768719, −9.60706481523003349664385670304, −7.67029392978510818125943826101, −6.93530905457497081719916784324, −6.149162717021580471777337938322, −5.37422080406047853023635507441, −4.60779482174091621225162673252, −3.61552721788206830596071858013, −2.236961171732689134598071476997, −1.10230222178921636210596157958, 0.55383636642888211409631279200, 1.89749100823023261188665361484, 2.87006677825738394330620730103, 3.93987095783878514216206250393, 5.09488709830651892545034544814, 5.86482568935181998202106194820, 6.51498896110083996387532249332, 7.0813652179872635424224247408, 9.013167103650291264569602767696, 10.09820407626287589510535129537, 10.45222335493941128961415942423, 11.61733717160385327351054396565, 12.19796661961130322626388130772, 13.0927643217944904998945356467, 13.73657469873681754788837150754, 14.856018111413753573655006472471, 15.45171506355249620432772215013, 16.72435465419500540258026484559, 16.95377171250569303550814589420, 18.265448004810454919296104467856, 19.082564220815700782513963595816, 19.69189946823846706097267749474, 21.07354196436673793594508041094, 21.76270347564666202162576736613, 22.245399459609082787073631717374

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