Properties

Label 1-709-709.7-r0-0-0
Degree $1$
Conductor $709$
Sign $0.791 - 0.610i$
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  + (0.895 + 0.445i)2-s + (−0.671 − 0.740i)3-s + (0.603 + 0.797i)4-s + (−0.589 − 0.807i)5-s + (−0.271 − 0.962i)6-s + (0.421 − 0.906i)7-s + (0.185 + 0.982i)8-s + (−0.0974 + 0.995i)9-s + (−0.167 − 0.985i)10-s + (0.254 + 0.967i)11-s + (0.185 − 0.982i)12-s + (0.574 − 0.818i)13-s + (0.781 − 0.624i)14-s + (−0.202 + 0.979i)15-s + (−0.271 + 0.962i)16-s + (0.710 + 0.703i)17-s + ⋯
L(s)  = 1  + (0.895 + 0.445i)2-s + (−0.671 − 0.740i)3-s + (0.603 + 0.797i)4-s + (−0.589 − 0.807i)5-s + (−0.271 − 0.962i)6-s + (0.421 − 0.906i)7-s + (0.185 + 0.982i)8-s + (−0.0974 + 0.995i)9-s + (−0.167 − 0.985i)10-s + (0.254 + 0.967i)11-s + (0.185 − 0.982i)12-s + (0.574 − 0.818i)13-s + (0.781 − 0.624i)14-s + (−0.202 + 0.979i)15-s + (−0.271 + 0.962i)16-s + (0.710 + 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.791 - 0.610i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.791 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.834072381 - 0.6251476728i\)
\(L(\frac12)\) \(\approx\) \(1.834072381 - 0.6251476728i\)
\(L(1)\) \(\approx\) \(1.440832986 - 0.1749951011i\)
\(L(1)\) \(\approx\) \(1.440832986 - 0.1749951011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.895 + 0.445i)T \)
3 \( 1 + (-0.671 - 0.740i)T \)
5 \( 1 + (-0.589 - 0.807i)T \)
7 \( 1 + (0.421 - 0.906i)T \)
11 \( 1 + (0.254 + 0.967i)T \)
13 \( 1 + (0.574 - 0.818i)T \)
17 \( 1 + (0.710 + 0.703i)T \)
19 \( 1 + (0.758 - 0.651i)T \)
23 \( 1 + (-0.981 - 0.194i)T \)
29 \( 1 + (0.999 - 0.0354i)T \)
31 \( 1 + (-0.746 - 0.665i)T \)
37 \( 1 + (0.421 - 0.906i)T \)
41 \( 1 + (-0.560 - 0.828i)T \)
43 \( 1 + (0.545 + 0.838i)T \)
47 \( 1 + (0.658 - 0.752i)T \)
53 \( 1 + (0.949 - 0.314i)T \)
59 \( 1 + (0.185 + 0.982i)T \)
61 \( 1 + (0.00887 - 0.999i)T \)
67 \( 1 + (0.0443 - 0.999i)T \)
71 \( 1 + (-0.167 + 0.985i)T \)
73 \( 1 + (-0.132 - 0.991i)T \)
79 \( 1 + (0.355 - 0.934i)T \)
83 \( 1 + (-0.339 - 0.940i)T \)
89 \( 1 + (-0.852 + 0.522i)T \)
97 \( 1 + (-0.973 - 0.228i)T \)
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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

−22.36259053619796722675708502904, −22.04342567348478006412704721026, −21.30974111661207020763294162983, −20.60608754922383533123517265239, −19.51731169237176186470305134905, −18.54055914386760192232914881094, −18.21727253877901663824744686087, −16.52215557117722616600132498052, −15.97557299568273722280888834744, −15.334475010418839551861792757861, −14.27333300949800373539134308460, −14.016754985372912008963497431568, −12.30068700143102366497687841912, −11.70411170637287802279197462451, −11.35681741609857444811897250210, −10.41370766106773128468238247627, −9.54976963193328891751723486727, −8.37864001415743538320429446679, −6.99272451397476609173291965064, −6.043028800800954438795460460, −5.49294829900613869100466121812, −4.363342362244950355579480988745, −3.53667831653535161379194550283, −2.78834993549952316272991914231, −1.236795445476925173355729531226, 0.89307778170403720159778778013, 2.00929383089478633092580143873, 3.62776527465967329780203713207, 4.44216496882973438226947853613, 5.25905290634975378379438468106, 6.10402487302121280575256886738, 7.32277756266821396448696733551, 7.652254131295098228362117207688, 8.54194236978458246415938997971, 10.238597374877046570509425904506, 11.1778691478410789767895408363, 11.99732202421258697174249436744, 12.59967028440786138693169629294, 13.31114196161131217030387795557, 14.10489109041603628079603614499, 15.14843571432187430711809058495, 16.060703810253098810249540196463, 16.72892790755095683475022817935, 17.48886870528718906982442945264, 18.07150993313499290617411272975, 19.66476455557182736430275785135, 20.10797199023120866457296182677, 20.87204293461250165855936600856, 21.984028357690654276233833886263, 22.92562124467900913341771994240

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