Properties

Label 1-287-287.157-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: $\chi_{287} (157, \cdot )$
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

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

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