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Properties

Label 1-547-547.301-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.368 - 0.929i$
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  + (−0.958 + 0.283i)2-s + (0.623 − 0.781i)3-s + (0.838 − 0.544i)4-s + (−0.332 + 0.942i)5-s + (−0.376 + 0.926i)6-s + (0.211 − 0.977i)7-s + (−0.650 + 0.759i)8-s + (−0.222 − 0.974i)9-s + (0.0517 − 0.998i)10-s + (0.278 − 0.960i)11-s + (0.0976 − 0.995i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (0.529 + 0.848i)15-s + (0.407 − 0.913i)16-s + (−0.857 − 0.514i)17-s + ⋯
L(s)  = 1  + (−0.958 + 0.283i)2-s + (0.623 − 0.781i)3-s + (0.838 − 0.544i)4-s + (−0.332 + 0.942i)5-s + (−0.376 + 0.926i)6-s + (0.211 − 0.977i)7-s + (−0.650 + 0.759i)8-s + (−0.222 − 0.974i)9-s + (0.0517 − 0.998i)10-s + (0.278 − 0.960i)11-s + (0.0976 − 0.995i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (0.529 + 0.848i)15-s + (0.407 − 0.913i)16-s + (−0.857 − 0.514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4750441485 - 0.6995859534i\)
\(L(\frac12)\) \(\approx\) \(0.4750441485 - 0.6995859534i\)
\(L(1)\) \(\approx\) \(0.7358518366 - 0.2643412401i\)
\(L(1)\) \(\approx\) \(0.7358518366 - 0.2643412401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.958 + 0.283i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.332 + 0.942i)T \)
7 \( 1 + (0.211 - 0.977i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (-0.857 - 0.514i)T \)
19 \( 1 + (-0.980 + 0.194i)T \)
23 \( 1 + (0.863 + 0.504i)T \)
29 \( 1 + (-0.928 + 0.370i)T \)
31 \( 1 + (-0.700 - 0.713i)T \)
37 \( 1 + (0.300 + 0.953i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.641 - 0.767i)T \)
47 \( 1 + (0.428 - 0.903i)T \)
53 \( 1 + (0.469 - 0.882i)T \)
59 \( 1 + (0.799 + 0.600i)T \)
61 \( 1 + (-0.0632 - 0.997i)T \)
67 \( 1 + (-0.910 + 0.413i)T \)
71 \( 1 + (-0.177 + 0.984i)T \)
73 \( 1 + (0.529 - 0.848i)T \)
79 \( 1 + (-0.985 - 0.171i)T \)
83 \( 1 + (-0.0402 + 0.999i)T \)
89 \( 1 + (-0.832 - 0.553i)T \)
97 \( 1 + (0.166 - 0.986i)T \)
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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

−23.93207546933374760208831968401, −22.566320242349166383837410451664, −21.48339160578794157269757798621, −21.01176931913536671663914536742, −20.288917093589947161974066784957, −19.50809303329717430163533500708, −18.84102774903685868927638962343, −17.68862134287257040877240807649, −16.88355997157513814450801758888, −16.03726971020326664784245071135, −15.371860354966543126627222586417, −14.72053777092381678644837049808, −13.09500590421652418471996940706, −12.44045977076150500496877154717, −11.33723489154743716892003710752, −10.66108647224811192947779186453, −9.28386059998502174707644747587, −8.99679247859341022864326524081, −8.38173467140017032288059839126, −7.27020008716707844198063089618, −5.92467170838946284324613920342, −4.582578153445226257514579052136, −3.839237735184030558209945982574, −2.39396489673888880619471602850, −1.659332779516054074449187923743, 0.549616471225119400242728368776, 1.76115691175000396851124991883, 2.96474457199955130560035346272, 3.81668878860170088355192011408, 5.834868318171156766844627669272, 6.74003690731030085158737201104, 7.29020309218721257690355403197, 8.20567963209538735916414121192, 8.871166564057040997749421523372, 10.09971923203179755004174419499, 11.129896900556166413042674650054, 11.41165500218005696421204960821, 13.10102967919401117333612711437, 13.82198947790966395634107789828, 14.77527098978381942454905082835, 15.35405668382277400282883763426, 16.56022398633516026686628980408, 17.39520209363496995118756909793, 18.30348093548403191716827816171, 18.79116424676846042151249174666, 19.61435779483628909545323344746, 20.25098037091069967922197873939, 21.08645953256121135102855075861, 22.50665398112450118867191852413, 23.58100957081096375234545031355

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