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Properties

Label 1-547-547.25-r0-0-0
Degree $1$
Conductor $547$
Sign $0.983 - 0.179i$
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  + (−0.558 − 0.829i)2-s + (−0.222 + 0.974i)3-s + (−0.376 + 0.926i)4-s + (−0.667 + 0.744i)5-s + (0.932 − 0.359i)6-s + (−0.976 − 0.216i)7-s + (0.978 − 0.205i)8-s + (−0.900 − 0.433i)9-s + (0.990 + 0.137i)10-s + (0.799 − 0.600i)11-s + (−0.819 − 0.572i)12-s + (−0.988 − 0.149i)13-s + (0.365 + 0.930i)14-s + (−0.577 − 0.816i)15-s + (−0.717 − 0.696i)16-s + (0.998 + 0.0460i)17-s + ⋯
L(s)  = 1  + (−0.558 − 0.829i)2-s + (−0.222 + 0.974i)3-s + (−0.376 + 0.926i)4-s + (−0.667 + 0.744i)5-s + (0.932 − 0.359i)6-s + (−0.976 − 0.216i)7-s + (0.978 − 0.205i)8-s + (−0.900 − 0.433i)9-s + (0.990 + 0.137i)10-s + (0.799 − 0.600i)11-s + (−0.819 − 0.572i)12-s + (−0.988 − 0.149i)13-s + (0.365 + 0.930i)14-s + (−0.577 − 0.816i)15-s + (−0.717 − 0.696i)16-s + (0.998 + 0.0460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.983 - 0.179i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5776588262 - 0.05234085933i\)
\(L(\frac12)\) \(\approx\) \(0.5776588262 - 0.05234085933i\)
\(L(1)\) \(\approx\) \(0.5800920979 + 0.008733681841i\)
\(L(1)\) \(\approx\) \(0.5800920979 + 0.008733681841i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.558 - 0.829i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.667 + 0.744i)T \)
7 \( 1 + (-0.976 - 0.216i)T \)
11 \( 1 + (0.799 - 0.600i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.998 + 0.0460i)T \)
19 \( 1 + (0.343 - 0.939i)T \)
23 \( 1 + (0.756 + 0.654i)T \)
29 \( 1 + (-0.999 - 0.0345i)T \)
31 \( 1 + (-0.479 - 0.877i)T \)
37 \( 1 + (-0.397 + 0.917i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.895 + 0.444i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (0.995 - 0.0919i)T \)
59 \( 1 + (0.948 + 0.316i)T \)
61 \( 1 + (0.863 + 0.504i)T \)
67 \( 1 + (0.905 - 0.423i)T \)
71 \( 1 + (0.605 - 0.795i)T \)
73 \( 1 + (-0.577 + 0.816i)T \)
79 \( 1 + (-0.832 + 0.553i)T \)
83 \( 1 + (0.692 + 0.721i)T \)
89 \( 1 + (-0.868 - 0.495i)T \)
97 \( 1 + (0.968 - 0.250i)T \)
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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

−23.38310268216787751865762718829, −22.99512768517326284606890647253, −22.21080306715230946940692099972, −20.46435521500147689860698254665, −19.65641228443560635677935905308, −19.15797027783310554766635030287, −18.45392840425907280359751144735, −17.29666119787767218140884938428, −16.68378384415532066232281724806, −16.176462347658029294160659970886, −14.867201674547257506639060491810, −14.29305383181591448053240679688, −12.94245349265747146113307664881, −12.42648116180838334038656545922, −11.570540662222524955396585775309, −10.11945402879711226801675374044, −9.254970513452335409044335815186, −8.44122076558792176927290718362, −7.359005259383523353280477244127, −6.97689838134672502487043214492, −5.789238450296873434208360863291, −5.019229096163857588734645124803, −3.61159020092094107837544511495, −1.91798576235428337581142005763, −0.763790136732610561823053067156, 0.60735898508018595268779094859, 2.69784218216008108701064840939, 3.41670746911340408820135759723, 4.02252898968678043854974527531, 5.36650127821947282296252791446, 6.7747056587269591539314646755, 7.63039988624331597754470284168, 8.92147943289527797135202666233, 9.65606787539431032642755069021, 10.29494526553026733654299307969, 11.317937747700073892037972801634, 11.73078373683876259740797681980, 12.843248019368162569828524965140, 14.02792599879705794968392467854, 14.97394813130957425373498516787, 15.90786620430147343968194727392, 16.78406003427713115822704581114, 17.27443628528651678332199224947, 18.58655101666270808874330923356, 19.417037107711242635339332771771, 19.77081811521411181888014055326, 20.81478615134654070677739780704, 21.813647854625157313882556457753, 22.41252284626293596598076851848, 22.75505695703653566938816567371

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