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Properties

Label 1-547-547.326-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: $\chi_{547} (326, \cdot )$
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

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

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