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Properties

Label 1-709-709.52-r1-0-0
Degree $1$
Conductor $709$
Sign $0.901 + 0.432i$
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  + (0.567 − 0.823i)2-s + (−0.992 − 0.123i)3-s + (−0.355 − 0.934i)4-s + (−0.895 + 0.445i)5-s + (−0.665 + 0.746i)6-s + (0.999 − 0.0354i)7-s + (−0.971 − 0.237i)8-s + (0.969 + 0.245i)9-s + (−0.141 + 0.989i)10-s + (0.589 − 0.807i)11-s + (0.237 + 0.971i)12-s + (0.847 + 0.530i)13-s + (0.537 − 0.842i)14-s + (0.943 − 0.330i)15-s + (−0.746 + 0.665i)16-s + (0.914 + 0.405i)17-s + ⋯
L(s)  = 1  + (0.567 − 0.823i)2-s + (−0.992 − 0.123i)3-s + (−0.355 − 0.934i)4-s + (−0.895 + 0.445i)5-s + (−0.665 + 0.746i)6-s + (0.999 − 0.0354i)7-s + (−0.971 − 0.237i)8-s + (0.969 + 0.245i)9-s + (−0.141 + 0.989i)10-s + (0.589 − 0.807i)11-s + (0.237 + 0.971i)12-s + (0.847 + 0.530i)13-s + (0.537 − 0.842i)14-s + (0.943 − 0.330i)15-s + (−0.746 + 0.665i)16-s + (0.914 + 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.901 + 0.432i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.901 + 0.432i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058501607 + 0.2409091508i\)
\(L(\frac12)\) \(\approx\) \(1.058501607 + 0.2409091508i\)
\(L(1)\) \(\approx\) \(0.8525728581 - 0.3505372370i\)
\(L(1)\) \(\approx\) \(0.8525728581 - 0.3505372370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.567 - 0.823i)T \)
3 \( 1 + (-0.992 - 0.123i)T \)
5 \( 1 + (-0.895 + 0.445i)T \)
7 \( 1 + (0.999 - 0.0354i)T \)
11 \( 1 + (0.589 - 0.807i)T \)
13 \( 1 + (0.847 + 0.530i)T \)
17 \( 1 + (0.914 + 0.405i)T \)
19 \( 1 + (-0.722 - 0.691i)T \)
23 \( 1 + (-0.476 + 0.879i)T \)
29 \( 1 + (-0.981 - 0.194i)T \)
31 \( 1 + (-0.651 + 0.758i)T \)
37 \( 1 + (0.0354 + 0.999i)T \)
41 \( 1 + (-0.610 - 0.792i)T \)
43 \( 1 + (-0.684 - 0.728i)T \)
47 \( 1 + (0.0266 + 0.999i)T \)
53 \( 1 + (-0.982 - 0.185i)T \)
59 \( 1 + (-0.237 + 0.971i)T \)
61 \( 1 + (0.740 + 0.671i)T \)
67 \( 1 + (0.515 - 0.857i)T \)
71 \( 1 + (-0.141 - 0.989i)T \)
73 \( 1 + (0.0532 + 0.998i)T \)
79 \( 1 + (0.347 - 0.937i)T \)
83 \( 1 + (-0.899 + 0.437i)T \)
89 \( 1 + (0.993 - 0.115i)T \)
97 \( 1 + (-0.297 + 0.954i)T \)
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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

−22.6578267980522865343134083521, −21.77766105358639553741971652875, −20.7519535791306135758157604009, −20.3857725982940562458641646308, −18.65249025982652721201576349493, −18.15040927313805222684517596982, −17.112572528639444573109909530550, −16.63360390132117729579264624176, −15.8500857982947777779289809566, −14.9560263653187594193505465329, −14.47497408002381746953555495626, −12.99411353289871620465040650994, −12.430711105052225131760381158311, −11.67031739747687710218509589795, −10.99844067457920615754292722482, −9.66599389329857959685032178295, −8.4333485037201763295652236883, −7.78558635087223126111833628393, −6.92433163698269150681356397509, −5.85641764425235813326970272819, −5.0929011600203322439241150494, −4.26974863008866228087198652824, −3.66062949849090886385151538061, −1.64195225801593739674234655265, −0.28669843521970246794281543845, 0.99877246664660020206108104563, 1.807826854815514874809169172510, 3.4819115702055074337465879192, 4.07055532237380684067155004907, 5.07106609547215849891400273022, 5.97337059702690350987496984860, 6.84838730959508067667399862226, 8.031872555923458210348505865116, 9.067291601757547936892305174606, 10.46361190768493441465602694204, 11.01400539887823828226900371242, 11.620476916505952043669786412422, 12.097417049479521247009355095, 13.26210297569473201634535842031, 14.115250812445317261324663012433, 14.972434863241470704993389733254, 15.77953012551699058996349052001, 16.8008246926929728113416084842, 17.7415923661431223585622535369, 18.75036446746765385534361169409, 18.98171379252990779481087551784, 20.07035805442980270906010745638, 21.11059741740961722030349668881, 21.70615539365673990632347230423, 22.37455302420251705269812820505

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