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Properties

Label 1-547-547.530-r0-0-0
Degree $1$
Conductor $547$
Sign $0.995 + 0.0986i$
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  + (0.166 − 0.986i)2-s + (0.623 + 0.781i)3-s + (−0.944 − 0.327i)4-s + (0.999 + 0.0230i)5-s + (0.874 − 0.484i)6-s + (0.995 − 0.0919i)7-s + (−0.479 + 0.877i)8-s + (−0.222 + 0.974i)9-s + (0.188 − 0.982i)10-s + (−0.632 + 0.774i)11-s + (−0.332 − 0.942i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (0.605 + 0.795i)15-s + (0.785 + 0.618i)16-s + (−0.0632 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.166 − 0.986i)2-s + (0.623 + 0.781i)3-s + (−0.944 − 0.327i)4-s + (0.999 + 0.0230i)5-s + (0.874 − 0.484i)6-s + (0.995 − 0.0919i)7-s + (−0.479 + 0.877i)8-s + (−0.222 + 0.974i)9-s + (0.188 − 0.982i)10-s + (−0.632 + 0.774i)11-s + (−0.332 − 0.942i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (0.605 + 0.795i)15-s + (0.785 + 0.618i)16-s + (−0.0632 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.995 + 0.0986i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (530, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.995 + 0.0986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.079603572 + 0.1028730632i\)
\(L(\frac12)\) \(\approx\) \(2.079603572 + 0.1028730632i\)
\(L(1)\) \(\approx\) \(1.546457866 - 0.1529271165i\)
\(L(1)\) \(\approx\) \(1.546457866 - 0.1529271165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.166 - 0.986i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.999 + 0.0230i)T \)
7 \( 1 + (0.995 - 0.0919i)T \)
11 \( 1 + (-0.632 + 0.774i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (-0.0632 + 0.997i)T \)
19 \( 1 + (-0.778 - 0.627i)T \)
23 \( 1 + (-0.980 - 0.194i)T \)
29 \( 1 + (0.940 - 0.338i)T \)
31 \( 1 + (-0.289 + 0.957i)T \)
37 \( 1 + (0.407 + 0.913i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.806 - 0.591i)T \)
47 \( 1 + (-0.919 - 0.391i)T \)
53 \( 1 + (-0.991 + 0.126i)T \)
59 \( 1 + (0.428 - 0.903i)T \)
61 \( 1 + (-0.998 + 0.0575i)T \)
67 \( 1 + (0.983 + 0.183i)T \)
71 \( 1 + (0.300 - 0.953i)T \)
73 \( 1 + (0.605 - 0.795i)T \)
79 \( 1 + (0.915 - 0.402i)T \)
83 \( 1 + (0.948 + 0.316i)T \)
89 \( 1 + (0.449 + 0.893i)T \)
97 \( 1 + (0.905 + 0.423i)T \)
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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

−23.64350056715476697239592717396, −22.83645262739792410345700055889, −21.527705941014613853380390579700, −21.096079043160611833081585840987, −20.12800451907172691857175052044, −18.631265428206439938807747713460, −18.22928692327089669376575039337, −17.69491371138808689624647402571, −16.64498940640595677655142776921, −15.66507337017321322677497783175, −14.5973122856888121121897365264, −14.07199776394616264540024248867, −13.37202772821742809071981903471, −12.719559815206442904180388728676, −11.43205976437514888442129904719, −10.130451981990865650587528407, −9.05503476660009413114914905742, −8.23310888485514773977397522174, −7.77032019348431101067354936641, −6.42406272666782866829355246994, −5.85710088975483301881718576696, −4.86759210037221204061469387512, −3.45997751845094568502924107745, −2.3075790998664656787915598450, −1.030283823987165805742758889734, 1.76716387139527537311550165479, 2.13432171175160724808642118669, 3.460219220177729423769278655316, 4.56274238833228692779815621895, 5.06450144295311544726325557767, 6.35289769831647415022591667607, 8.17082460155436434764976134954, 8.687657410457678855116947783, 9.72768051795263693523633232912, 10.48338038304206027688415125418, 10.961559920476990089008774920222, 12.24653120445127413178540334420, 13.34233167799150147276652559630, 13.89993703246426699612390309932, 14.68901698693944460430795681434, 15.45363637059922801274282619781, 16.87442967073460879347305693486, 17.694798953832289002673933925036, 18.37285690022536947689121717149, 19.44824705016199433432423960532, 20.409918745640378125482404708029, 20.88337267661557929478838183322, 21.60076067626698653733147507376, 21.98985661637969060962693223787, 23.32857299743215680040178326483

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