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Properties

Label 1-287-287.74-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.0595 + 0.998i$
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  + (0.978 + 0.207i)2-s + (−0.866 + 0.5i)3-s + (0.913 + 0.406i)4-s + (0.104 + 0.994i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)9-s + (−0.104 + 0.994i)10-s + (0.994 + 0.104i)11-s + (−0.994 + 0.104i)12-s + (0.951 − 0.309i)13-s + (−0.587 − 0.809i)15-s + (0.669 + 0.743i)16-s + (−0.994 − 0.104i)17-s + (0.669 − 0.743i)18-s + (−0.743 + 0.669i)19-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (−0.866 + 0.5i)3-s + (0.913 + 0.406i)4-s + (0.104 + 0.994i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)9-s + (−0.104 + 0.994i)10-s + (0.994 + 0.104i)11-s + (−0.994 + 0.104i)12-s + (0.951 − 0.309i)13-s + (−0.587 − 0.809i)15-s + (0.669 + 0.743i)16-s + (−0.994 − 0.104i)17-s + (0.669 − 0.743i)18-s + (−0.743 + 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0595 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0595 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0595 + 0.998i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.0595 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266976142 + 1.344838916i\)
\(L(\frac12)\) \(\approx\) \(1.266976142 + 1.344838916i\)
\(L(1)\) \(\approx\) \(1.360154796 + 0.7371429002i\)
\(L(1)\) \(\approx\) \(1.360154796 + 0.7371429002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (-0.994 - 0.104i)T \)
19 \( 1 + (-0.743 + 0.669i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (0.743 - 0.669i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−24.855502534025226525153684521726, −24.146990440987064279932885320320, −23.62768538704494638530874791411, −22.58016793945215744476004305371, −21.845597374257640772862863096970, −20.98004921554123471038736489129, −19.902440602004753318714075927539, −19.21134562532590227954042484588, −17.826935621536864804856436311948, −16.88590354607916757135196398561, −16.156397540233902523683634616222, −15.21079222937484973031692639443, −13.59777603233446499602234133675, −13.35519064276911307596228431407, −12.15369168936065179061679734639, −11.60448530017366982889679211646, −10.65086489974378147872529827947, −9.24847502098457785884358543297, −7.92147430725518113269197747748, −6.41586814728402150992378110504, −6.087551341476991685198596596789, −4.68223057292078842815272722623, −4.09145129392516712229690639617, −2.120644739394643336062474721531, −1.102824667632306121562886538923, 1.887739579297369648619460049285, 3.51116625907695372132499842185, 4.14352886237126689323996994188, 5.51359627844767088524331834970, 6.43448092466569278035807201349, 6.9286464801658168053006664321, 8.59951567355123348191209415247, 10.222643592104601132234733797829, 10.9027939048242498243677496762, 11.721621227514660075339334368595, 12.643652639407427759915554937010, 13.909517814912306227005474204062, 14.69823395990624585189181649775, 15.59440121328655201724798815504, 16.34216444011895677448378452790, 17.44760865262649463176815268556, 18.19971086786776897481185414348, 19.60765152440441596507531390391, 20.64622589565503106701548150619, 21.74411005235630411851912030769, 22.12385802961977727442248659395, 23.034824112559840286586493658337, 23.52523597181143976707108863753, 24.78069707667095140230480635137, 25.650253500680439902979330853

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