Properties

Label 1-287-287.117-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.322 - 0.946i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (−0.406 + 0.913i)2-s + (−0.258 − 0.965i)3-s + (−0.669 − 0.743i)4-s + (−0.207 − 0.978i)5-s + (0.987 + 0.156i)6-s + (0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (0.978 + 0.207i)10-s + (0.838 − 0.544i)11-s + (−0.544 + 0.838i)12-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (−0.104 + 0.994i)16-s + (0.544 + 0.838i)17-s + (−0.104 − 0.994i)18-s + (0.777 − 0.629i)19-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.258 − 0.965i)3-s + (−0.669 − 0.743i)4-s + (−0.207 − 0.978i)5-s + (0.987 + 0.156i)6-s + (0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (0.978 + 0.207i)10-s + (0.838 − 0.544i)11-s + (−0.544 + 0.838i)12-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (−0.104 + 0.994i)16-s + (0.544 + 0.838i)17-s + (−0.104 − 0.994i)18-s + (0.777 − 0.629i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.322 - 0.946i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.322 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3873711111 - 0.5410447269i\)
\(L(\frac12)\) \(\approx\) \(0.3873711111 - 0.5410447269i\)
\(L(1)\) \(\approx\) \(0.6693794573 - 0.1983626922i\)
\(L(1)\) \(\approx\) \(0.6693794573 - 0.1983626922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (-0.838 + 0.544i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (-0.544 - 0.838i)T \)
19 \( 1 + (-0.777 + 0.629i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.358 + 0.933i)T \)
53 \( 1 + (0.998 + 0.0523i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.994 - 0.104i)T \)
67 \( 1 + (0.0523 - 0.998i)T \)
71 \( 1 + (-0.891 - 0.453i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.629 + 0.777i)T \)
97 \( 1 + (-0.891 + 0.453i)T \)
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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

−26.08608105691289323798061565850, −25.416161814387869077158731850909, −23.585054243427913609274256318159, −22.60094783464956352274177053545, −22.207767870224428493271137552657, −21.296667443867738520851589112643, −20.38588516125644116413796202475, −19.57884494230489642286042296459, −18.50629217686039689077660869323, −17.788479897562076182467600584569, −16.67709747545158892901039880759, −15.916866691269617338498162529367, −14.41712866822729526238465379302, −14.10007411679332230415945353875, −12.23468867509471775002825230126, −11.5724072742216635062834393532, −10.8328605399031420057213195195, −9.71894629579921831171793747595, −9.32014212303135575759828725290, −7.82991428043872915196492029788, −6.655434273636345987245177599421, −5.11435395813212684676612349716, −3.86990918715958967182090340611, −3.29201455503652374428456056488, −1.78024193698733302593674991167, 0.55732238958747457400700855071, 1.63581439905950731814913906190, 3.78799030110602445383523264432, 5.3188778603209642251729285460, 5.90687085388655249364266265957, 7.09457873334791058452467431706, 8.128453932520971155145333446786, 8.64468080818102809707921982593, 9.90198014396063498227540035518, 11.28229480906045229250828097380, 12.376660449225233813192063663739, 13.270464701137680385152754188886, 14.07471417240278606506752440275, 15.25130978065938243286060982201, 16.343238355197021420952691367521, 17.0408436223825307012162334188, 17.74278178106052511385706734541, 18.75779227457630759568662282586, 19.630135546794248826653486712427, 20.30399142578957690247320295458, 22.0272556654626816084738452797, 22.85348155719568573799827179244, 23.900028498011065473831765266746, 24.32098591240871813385079347311, 25.06444075160990993874955023139

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