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Properties

Label 1-712-712.581-r0-0-0
Degree $1$
Conductor $712$
Sign $0.989 + 0.144i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (0.755 + 0.654i)3-s + (−0.959 + 0.281i)5-s + (−0.281 − 0.959i)7-s + (0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.755 − 0.654i)13-s + (−0.909 − 0.415i)15-s + (−0.415 − 0.909i)17-s + (0.989 − 0.142i)19-s + (0.415 − 0.909i)21-s + (0.989 − 0.142i)23-s + (0.841 − 0.540i)25-s + (−0.540 + 0.841i)27-s + (−0.281 − 0.959i)29-s + (0.989 + 0.142i)31-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (−0.959 + 0.281i)5-s + (−0.281 − 0.959i)7-s + (0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.755 − 0.654i)13-s + (−0.909 − 0.415i)15-s + (−0.415 − 0.909i)17-s + (0.989 − 0.142i)19-s + (0.415 − 0.909i)21-s + (0.989 − 0.142i)23-s + (0.841 − 0.540i)25-s + (−0.540 + 0.841i)27-s + (−0.281 − 0.959i)29-s + (0.989 + 0.142i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.554504906 + 0.1128831162i\)
\(L(\frac12)\) \(\approx\) \(1.554504906 + 0.1128831162i\)
\(L(1)\) \(\approx\) \(1.190778961 + 0.1334205797i\)
\(L(1)\) \(\approx\) \(1.190778961 + 0.1334205797i\)

Euler product

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

−22.567135125707680681888209298843, −21.78821133066914752355996797554, −20.842333969663929823016246573845, −19.81062907107229397818714901147, −19.37241454379646282628178498007, −18.8561118970559646209847572499, −17.84687002417911684763879156379, −16.81669929964089487201018615777, −15.91144968087434901093363469573, −14.99043004024447393298513791973, −14.55202727726364170797609331962, −13.42899174308664282716537101271, −12.4564909566813689752498742778, −12.02092102455546120049342240307, −11.20117579909107728515977195593, −9.58947790405599800627238909681, −8.93569707995185496624818191217, −8.32314431071664564774185879116, −7.237712942131183292908122566784, −6.602902047923392028566069073999, −5.365332406308560496085532534681, −4.08407212019964530432226092163, −3.29909298778914338731855742448, −2.24236953072645524858764944321, −1.06866287607608044008429454762, 0.87674550076354691955128045720, 2.674165496653769380835067275, 3.37241392496353384829971529783, 4.31196538656026790405761263887, 4.93539732607838350354233352405, 6.69009045711360623947390230624, 7.430047029721881878226904856956, 8.09112368031041543525600801420, 9.33113642880477476858199968466, 9.867940901857278992412162398135, 10.909505113276731535570797730273, 11.62380902252704851662873193228, 12.73519859860252892421346774398, 13.79063778955239000808619428219, 14.39653006481266160710310883122, 15.3042061695144533070752163414, 15.85963295695699753027826083049, 16.79808935110059913865867254218, 17.55867717362978219063511592166, 18.97241310623675816573355192280, 19.464206306937313521874910795795, 20.345942650968763490726591514279, 20.51983270653732260603649326903, 22.01448154974148306150062347900, 22.60451181830944824209005734385

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