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Properties

Label 1-712-712.11-r1-0-0
Degree $1$
Conductor $712$
Sign $0.904 - 0.426i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.959 − 0.281i)3-s + (−0.415 + 0.909i)5-s + (0.415 − 0.909i)7-s + (0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.959 + 0.281i)13-s + (−0.142 + 0.989i)15-s + (−0.142 − 0.989i)17-s + (−0.841 + 0.540i)19-s + (0.142 − 0.989i)21-s + (0.841 − 0.540i)23-s + (−0.654 − 0.755i)25-s + (0.654 − 0.755i)27-s + (0.415 − 0.909i)29-s + (0.841 + 0.540i)31-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)3-s + (−0.415 + 0.909i)5-s + (0.415 − 0.909i)7-s + (0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.959 + 0.281i)13-s + (−0.142 + 0.989i)15-s + (−0.142 − 0.989i)17-s + (−0.841 + 0.540i)19-s + (0.142 − 0.989i)21-s + (0.841 − 0.540i)23-s + (−0.654 − 0.755i)25-s + (0.654 − 0.755i)27-s + (0.415 − 0.909i)29-s + (0.841 + 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.904 - 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.927675671 - 0.6550397055i\)
\(L(\frac12)\) \(\approx\) \(2.927675671 - 0.6550397055i\)
\(L(1)\) \(\approx\) \(1.509767635 - 0.1058094347i\)
\(L(1)\) \(\approx\) \(1.509767635 - 0.1058094347i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−22.066127240019804248117791938117, −21.50571253507262483585621912414, −20.97752559341007137206194544536, −19.71802813167169526301768190458, −19.554684399851729585343545939290, −18.67086019852120021418150737816, −17.41069998115966732971908143879, −16.68830137407532864305001382514, −15.68368349407935461288607443366, −15.07540162664192985559838081840, −14.421644287329876829204891756814, −13.22917059553728183315868557872, −12.668132617023770613967253084270, −11.69549982860643269677030460160, −10.74557293532802189506355388905, −9.522079171353182865559963547515, −8.79173748254730213188928888721, −8.34156463426972031120120481381, −7.43097770635281710926226893794, −6.03003666568468803858963158654, −4.95793287689138624658335814004, −4.240529905031170667518503270888, −3.10511705889198619583596503275, −2.14563772368828694156338507363, −0.95615646520464397694070393519, 0.73841643809188202799503552899, 2.13120355219734356551826757279, 2.8353847071616599258713029410, 4.1141047571663211135854808919, 4.564461463420288343782760728042, 6.48455216377539844100636220856, 7.21207308298788989680019817372, 7.63277211587132987711095704150, 8.74251132012350363188567435942, 9.850659935758132048923736240, 10.396834924338951530772131999877, 11.59170630322932750813303795460, 12.35956300753480419178635357177, 13.45797470169801148285297804105, 14.27480642531285046469114833754, 14.75259622326248858170299734767, 15.434452849023043857834564127759, 16.68186419760467437827900381360, 17.594732631032271361493995192446, 18.36037516860011682415990686409, 19.32473901685543021577079312691, 19.7319737318140715383607992856, 20.6456159298198235200625111000, 21.312838979504278108971307481504, 22.53005816487207266082333745706

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