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Properties

Label 1-709-709.363-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.660 + 0.750i$
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  + (0.658 + 0.752i)2-s + (−0.437 + 0.899i)3-s + (−0.132 + 0.991i)4-s + (−0.530 − 0.847i)5-s + (−0.964 + 0.263i)6-s + (0.949 + 0.314i)7-s + (−0.833 + 0.552i)8-s + (−0.617 − 0.786i)9-s + (0.288 − 0.957i)10-s + (0.574 − 0.818i)11-s + (−0.833 − 0.552i)12-s + (0.949 − 0.314i)13-s + (0.388 + 0.921i)14-s + (0.994 − 0.106i)15-s + (−0.964 − 0.263i)16-s + (0.574 + 0.818i)17-s + ⋯
L(s)  = 1  + (0.658 + 0.752i)2-s + (−0.437 + 0.899i)3-s + (−0.132 + 0.991i)4-s + (−0.530 − 0.847i)5-s + (−0.964 + 0.263i)6-s + (0.949 + 0.314i)7-s + (−0.833 + 0.552i)8-s + (−0.617 − 0.786i)9-s + (0.288 − 0.957i)10-s + (0.574 − 0.818i)11-s + (−0.833 − 0.552i)12-s + (0.949 − 0.314i)13-s + (0.388 + 0.921i)14-s + (0.994 − 0.106i)15-s + (−0.964 − 0.263i)16-s + (0.574 + 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6962596808 + 1.541187479i\)
\(L(\frac12)\) \(\approx\) \(0.6962596808 + 1.541187479i\)
\(L(1)\) \(\approx\) \(0.9896185844 + 0.8462357568i\)
\(L(1)\) \(\approx\) \(0.9896185844 + 0.8462357568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.658 + 0.752i)T \)
3 \( 1 + (-0.437 + 0.899i)T \)
5 \( 1 + (-0.530 - 0.847i)T \)
7 \( 1 + (0.949 + 0.314i)T \)
11 \( 1 + (0.574 - 0.818i)T \)
13 \( 1 + (0.949 - 0.314i)T \)
17 \( 1 + (0.574 + 0.818i)T \)
19 \( 1 + (-0.833 + 0.552i)T \)
23 \( 1 + (-0.237 + 0.971i)T \)
29 \( 1 + (0.185 + 0.982i)T \)
31 \( 1 + (0.994 - 0.106i)T \)
37 \( 1 + (0.949 + 0.314i)T \)
41 \( 1 + (-0.931 + 0.364i)T \)
43 \( 1 + (0.484 - 0.874i)T \)
47 \( 1 + (-0.237 + 0.971i)T \)
53 \( 1 + (0.994 - 0.106i)T \)
59 \( 1 + (-0.833 + 0.552i)T \)
61 \( 1 + (-0.339 - 0.940i)T \)
67 \( 1 + (-0.987 + 0.159i)T \)
71 \( 1 + (0.288 + 0.957i)T \)
73 \( 1 + (-0.887 - 0.461i)T \)
79 \( 1 + (-0.998 + 0.0532i)T \)
83 \( 1 + (0.802 - 0.596i)T \)
89 \( 1 + (0.861 - 0.507i)T \)
97 \( 1 + (0.910 + 0.413i)T \)
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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

−22.59297697200921426400740656364, −21.55033173766101650193476233585, −20.67407492125865452184930824659, −19.84611308989002051181563159890, −19.10461641990545134424686753096, −18.31087200055318737847684712446, −17.85995473987803849699808946094, −16.707350302181136933659632124343, −15.42071206152027607061425047044, −14.60377761724575221188904496879, −13.977652912111876903288297868122, −13.21717865689429707598996886137, −11.98307647019024700411068255264, −11.71847862700977810684637195386, −10.891153692396703358499698912163, −10.1822185180936678394947125629, −8.716905229267854606910381942448, −7.666270547659423296452374598, −6.73261321009893846967049863282, −6.105737997717330830598017494009, −4.75155382814652217082899743266, −4.11271404662856188761523885578, −2.74675088417638082431497111834, −1.91610060433993674994561125617, −0.79880860791896891232984170033, 1.28477865479195340734015065597, 3.30063518434548046754681340645, 3.997444625380439016895268920028, 4.75597658658545068897126524362, 5.69508357484821935545037638637, 6.18341344551470373195410579283, 7.82460350672057621128159072015, 8.50331373046935885975121812489, 9.01726825779315027836462212044, 10.540914349307311109811155880075, 11.52976985502008781008215595618, 11.976543331210618917134999712220, 13.00003203370898138970868222975, 14.06256686695461910965277117075, 14.88380346509712554870052481372, 15.53227693387953593761603584817, 16.28119802376476653502649093847, 16.96949651350932675043355044330, 17.53077838244078126924611175297, 18.69013702133466441041906415401, 20.00424337401124371809345228280, 20.87944492826401882555962136390, 21.38203884161266097858704585509, 21.98061929763386986276281422847, 23.18757384733114795391460541766

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