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Properties

Label 1-547-547.35-r0-0-0
Degree $1$
Conductor $547$
Sign $0.354 + 0.935i$
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  + (−0.994 − 0.103i)2-s + (−0.900 − 0.433i)3-s + (0.978 + 0.205i)4-s + (−0.952 + 0.305i)5-s + (0.851 + 0.524i)6-s + (0.322 + 0.946i)7-s + (−0.952 − 0.305i)8-s + (0.623 + 0.781i)9-s + (0.978 − 0.205i)10-s + (0.568 + 0.822i)11-s + (−0.792 − 0.609i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.990 + 0.137i)15-s + (0.915 + 0.402i)16-s + (0.997 − 0.0689i)17-s + ⋯
L(s)  = 1  + (−0.994 − 0.103i)2-s + (−0.900 − 0.433i)3-s + (0.978 + 0.205i)4-s + (−0.952 + 0.305i)5-s + (0.851 + 0.524i)6-s + (0.322 + 0.946i)7-s + (−0.952 − 0.305i)8-s + (0.623 + 0.781i)9-s + (0.978 − 0.205i)10-s + (0.568 + 0.822i)11-s + (−0.792 − 0.609i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.990 + 0.137i)15-s + (0.915 + 0.402i)16-s + (0.997 − 0.0689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.354 + 0.935i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.354 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3884288815 + 0.2681932030i\)
\(L(\frac12)\) \(\approx\) \(0.3884288815 + 0.2681932030i\)
\(L(1)\) \(\approx\) \(0.4852312501 + 0.05428142749i\)
\(L(1)\) \(\approx\) \(0.4852312501 + 0.05428142749i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.994 - 0.103i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.952 + 0.305i)T \)
7 \( 1 + (0.322 + 0.946i)T \)
11 \( 1 + (0.568 + 0.822i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.997 - 0.0689i)T \)
19 \( 1 + (0.256 - 0.966i)T \)
23 \( 1 + (-0.479 + 0.877i)T \)
29 \( 1 + (0.0517 + 0.998i)T \)
31 \( 1 + (-0.999 + 0.0345i)T \)
37 \( 1 + (-0.985 - 0.171i)T \)
41 \( 1 + T \)
43 \( 1 + (0.770 - 0.636i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (0.990 + 0.137i)T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (-0.700 + 0.713i)T \)
67 \( 1 + (-0.792 - 0.609i)T \)
71 \( 1 + (0.188 + 0.982i)T \)
73 \( 1 + (0.990 - 0.137i)T \)
79 \( 1 + (0.770 - 0.636i)T \)
83 \( 1 + (-0.354 + 0.935i)T \)
89 \( 1 + (-0.700 - 0.713i)T \)
97 \( 1 + (-0.928 - 0.370i)T \)
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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

−23.35544127991352011710514745949, −22.57919461678576854416527231856, −21.19704384487840779180545298908, −20.78287543698914084256924318357, −19.68244624341234178569876694026, −19.00276476648764622431935264053, −18.150785814036330479401370052946, −17.01731501475661262872298476901, −16.46918722943805164974957117221, −16.248255023643463888085453025809, −14.92620983790284382222980378967, −14.160672850681127457776234025107, −12.4065767513684937583165014072, −11.75643399366021874453997983091, −11.100249087565821983357054737877, −10.28949400275327495653479460277, −9.38881390570012494997856038160, −8.30559079529476538670109562145, −7.462105771316197998976603268388, −6.58830527830315566222845511117, −5.57283920518004634421557591126, −4.24813733326976766496965674944, −3.53650942143295341493857382293, −1.48918629756982303472822679164, −0.50492053023815453866664147324, 1.04928647103227073843068060300, 2.24977732893199075929416325749, 3.46452746662650680561257830249, 5.0367455244732974211666220841, 5.938024777598231317813560329830, 7.22389366866154208835877851057, 7.51524622756319466089009385868, 8.639382751620787825511968023410, 9.71403936859468072531877880666, 10.73978405380620213100917898462, 11.45359726976268942960431589341, 12.2232144600646799704456534849, 12.61647098357506198564698046379, 14.52549230989874488737031851241, 15.42157710727663601090521197443, 15.97082917840260588579034915734, 17.006635153684046664351847321440, 17.98749483477715619165539433908, 18.17204300716164411802212248796, 19.36566320578510778041415643135, 19.74148346572235218817264038517, 20.93544814125166784077784523822, 22.02898172537432556695931028502, 22.62898703818120788061028429840, 23.73360037015646737287696355618

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