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Properties

Label 1-547-547.99-r0-0-0
Degree $1$
Conductor $547$
Sign $0.796 + 0.604i$
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  + (−0.311 − 0.950i)2-s + (0.623 + 0.781i)3-s + (−0.806 + 0.591i)4-s + (0.0976 + 0.995i)5-s + (0.548 − 0.835i)6-s + (0.924 + 0.381i)7-s + (0.813 + 0.582i)8-s + (−0.222 + 0.974i)9-s + (0.915 − 0.402i)10-s + (0.987 − 0.160i)11-s + (−0.965 − 0.261i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (−0.717 + 0.696i)15-s + (0.300 − 0.953i)16-s + (−0.614 − 0.788i)17-s + ⋯
L(s)  = 1  + (−0.311 − 0.950i)2-s + (0.623 + 0.781i)3-s + (−0.806 + 0.591i)4-s + (0.0976 + 0.995i)5-s + (0.548 − 0.835i)6-s + (0.924 + 0.381i)7-s + (0.813 + 0.582i)8-s + (−0.222 + 0.974i)9-s + (0.915 − 0.402i)10-s + (0.987 − 0.160i)11-s + (−0.965 − 0.261i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (−0.717 + 0.696i)15-s + (0.300 − 0.953i)16-s + (−0.614 − 0.788i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.579432562 + 0.5315631549i\)
\(L(\frac12)\) \(\approx\) \(1.579432562 + 0.5315631549i\)
\(L(1)\) \(\approx\) \(1.251759165 + 0.1265103843i\)
\(L(1)\) \(\approx\) \(1.251759165 + 0.1265103843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.311 - 0.950i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.0976 + 0.995i)T \)
7 \( 1 + (0.924 + 0.381i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (-0.614 - 0.788i)T \)
19 \( 1 + (0.863 - 0.504i)T \)
23 \( 1 + (0.998 - 0.0460i)T \)
29 \( 1 + (-0.994 + 0.103i)T \)
31 \( 1 + (0.997 - 0.0689i)T \)
37 \( 1 + (-0.177 - 0.984i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.944 + 0.327i)T \)
47 \( 1 + (0.799 + 0.600i)T \)
53 \( 1 + (-0.244 - 0.969i)T \)
59 \( 1 + (-0.996 - 0.0804i)T \)
61 \( 1 + (-0.857 + 0.514i)T \)
67 \( 1 + (0.709 - 0.705i)T \)
71 \( 1 + (0.785 + 0.618i)T \)
73 \( 1 + (-0.717 - 0.696i)T \)
79 \( 1 + (0.188 - 0.982i)T \)
83 \( 1 + (-0.200 + 0.979i)T \)
89 \( 1 + (-0.0172 + 0.999i)T \)
97 \( 1 + (-0.958 + 0.283i)T \)
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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

−23.57350702458223015404869110166, −22.92941236467548030945688459082, −21.58507976609357551900097893344, −20.341502988182462532842724945459, −20.105157784643924408803545405220, −18.91478155037951983542527885870, −18.14293774309649582302541284056, −17.19150475167380885061314727567, −16.944200010852402528822980721563, −15.48252895457466623801775362699, −14.90584519817664178512278539959, −13.80855927097894962224698031123, −13.430439375968166635702427836046, −12.397637372589587193689705321413, −11.299972270559855139099814532651, −9.90711559000279303038272357706, −8.8947840060236512867798668153, −8.38887715669119160585908911340, −7.65427540184108556175621653248, −6.62816969142176835395760474353, −5.70122614528013415174417502255, −4.59828668851777311250042084138, −3.633178881516728264377290072, −1.53896532654288859669915798060, −1.143128391938760944591595871848, 1.55762167860662401898046180486, 2.54876233215337781354423431250, 3.43108047608696589804805495232, 4.306647360349635368628719073593, 5.326600371438567358358083253778, 6.92968692362642652817574558792, 7.99458272884640896760059680939, 9.08635524889018894141707145953, 9.3698276146992306982573022408, 10.730067877247002102944679063256, 11.20315509505161691086662409716, 11.83975543700524027731240184285, 13.54962896130353666885577224031, 13.98020038950277015584956502387, 14.8037903937731183216788184192, 15.732348714096935814715837546986, 16.90767354698837673771261559317, 17.8106116638063549931137669272, 18.62917346968015979173527359394, 19.28691458919897331886345874060, 20.25882425695415423504489730440, 20.94887797244374594711247613172, 21.67129204512899504322616520678, 22.30845875341214589973302744153, 22.98282650007713669762751067427

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