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Properties

Label 1-547-547.502-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.760 + 0.649i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (−0.806 + 0.591i)2-s + (−0.222 + 0.974i)3-s + (0.300 − 0.953i)4-s + (−0.980 + 0.194i)5-s + (−0.397 − 0.917i)6-s + (0.709 + 0.705i)7-s + (0.322 + 0.946i)8-s + (−0.900 − 0.433i)9-s + (0.675 − 0.736i)10-s + (0.948 − 0.316i)11-s + (0.863 + 0.504i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.0287 − 0.999i)15-s + (−0.819 − 0.572i)16-s + (−0.244 + 0.969i)17-s + ⋯
L(s)  = 1  + (−0.806 + 0.591i)2-s + (−0.222 + 0.974i)3-s + (0.300 − 0.953i)4-s + (−0.980 + 0.194i)5-s + (−0.397 − 0.917i)6-s + (0.709 + 0.705i)7-s + (0.322 + 0.946i)8-s + (−0.900 − 0.433i)9-s + (0.675 − 0.736i)10-s + (0.948 − 0.316i)11-s + (0.863 + 0.504i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.0287 − 0.999i)15-s + (−0.819 − 0.572i)16-s + (−0.244 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.760 + 0.649i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (502, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.760 + 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2771797961 + 0.7516010545i\)
\(L(\frac12)\) \(\approx\) \(0.2771797961 + 0.7516010545i\)
\(L(1)\) \(\approx\) \(0.5288773487 + 0.4503324367i\)
\(L(1)\) \(\approx\) \(0.5288773487 + 0.4503324367i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.806 + 0.591i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.980 + 0.194i)T \)
7 \( 1 + (0.709 + 0.705i)T \)
11 \( 1 + (0.948 - 0.316i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.244 + 0.969i)T \)
19 \( 1 + (0.490 - 0.871i)T \)
23 \( 1 + (0.995 - 0.0919i)T \)
29 \( 1 + (0.978 - 0.205i)T \)
31 \( 1 + (0.990 - 0.137i)T \)
37 \( 1 + (-0.937 + 0.349i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.785 - 0.618i)T \)
47 \( 1 + (0.278 + 0.960i)T \)
53 \( 1 + (-0.880 + 0.474i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (0.469 - 0.882i)T \)
67 \( 1 + (0.00575 - 0.999i)T \)
71 \( 1 + (0.233 + 0.972i)T \)
73 \( 1 + (0.0287 + 0.999i)T \)
79 \( 1 + (-0.928 - 0.370i)T \)
83 \( 1 + (-0.919 - 0.391i)T \)
89 \( 1 + (-0.999 - 0.0345i)T \)
97 \( 1 + (0.838 - 0.544i)T \)
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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

−22.77676888152113276677677877512, −22.67257117585541665461698872636, −20.88599493304432188678792911434, −20.35334348216029018768113377539, −19.61917720593406280036202977460, −18.981449762818038327008130743703, −17.97309303009090762128216152193, −17.44803662985777872635744021487, −16.616633145371206342811688885979, −15.68908536513431028867483941757, −14.393245595926876369226134408471, −13.44338439277017561534786618603, −12.40599904862777090683685793005, −11.80851283573145765747338415043, −11.15697352676113083475454712737, −10.25328989528340745256251164905, −8.8224129438116835261036472381, −8.169471327517736044882354111, −7.34375249798736182537750830725, −6.79026582247785167583918614472, −5.087980776442691559519670676611, −3.89020109063957399330165639880, −2.87920358120471453572321723482, −1.40152302523556521214867756509, −0.74804122474759371987619325378, 1.18807487499262348898523664554, 2.82912988836116861961651919547, 4.20672596208476562210972974429, 4.9281987171114673839843209995, 6.15253212019524086161134436997, 6.92646856323564272003497041159, 8.38823347710272910959339487855, 8.68837667971642575973173997233, 9.62149439908724301278173585985, 10.89536275382714007364449157201, 11.33805038104974810634692390569, 12.04535600958551014828881071460, 14.04485818781524564572583080526, 14.68573507579336176712799729598, 15.54137457843890841803368288999, 15.880465616822233212392004188858, 17.05301369425989760107384036719, 17.49274981050745256457998024344, 18.78660126107718193707802863641, 19.32637314923035134247863718094, 20.22516403608339922916449071102, 21.19971822948904555065022371420, 22.05387777545647028822640937713, 22.973751000963973482094688212020, 23.85530104383372783132372519574

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