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Properties

Label 1-287-287.143-r1-0-0
Degree $1$
Conductor $287$
Sign $0.564 - 0.825i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (−0.913 + 0.406i)2-s + (0.866 − 0.5i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (−0.587 + 0.809i)6-s + (−0.309 + 0.951i)8-s + (0.5 − 0.866i)9-s + (0.978 − 0.207i)10-s + (0.207 + 0.978i)11-s + (0.207 − 0.978i)12-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)15-s + (−0.104 − 0.994i)16-s + (0.207 + 0.978i)17-s + (−0.104 + 0.994i)18-s + (−0.994 + 0.104i)19-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.866 − 0.5i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (−0.587 + 0.809i)6-s + (−0.309 + 0.951i)8-s + (0.5 − 0.866i)9-s + (0.978 − 0.207i)10-s + (0.207 + 0.978i)11-s + (0.207 − 0.978i)12-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)15-s + (−0.104 − 0.994i)16-s + (0.207 + 0.978i)17-s + (−0.104 + 0.994i)18-s + (−0.994 + 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229035120 - 0.6480460680i\)
\(L(\frac12)\) \(\approx\) \(1.229035120 - 0.6480460680i\)
\(L(1)\) \(\approx\) \(0.8804690631 - 0.1198621920i\)
\(L(1)\) \(\approx\) \(0.8804690631 - 0.1198621920i\)

Euler product

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

−25.74081032787762790056883644999, −24.83415227483080824984640391038, −23.831300333746445283189906266002, −22.5428093608678849920803247755, −21.358157936324840654417088147, −20.89962043382675686769520147789, −19.76633384154718351547215311728, −19.10795786732929243375265334724, −18.67188236470581011621492146566, −17.13272293228236431137025936043, −16.06524787298542867282201612978, −15.712254565047760956215005841512, −14.44812055496147677874674104206, −13.41131695136558884895298864424, −12.047732980071098963493423578638, −11.15285471126110247599225502402, −10.41563174495753813587495717502, −9.034391288858510996425001117608, −8.619574402667526135666646377983, −7.586917114228214553900087487101, −6.600167751146054432002143381340, −4.511418755036088829449887886939, −3.47844195916780527208098456687, −2.694528028643612761334961698357, −1.05749982892085705394979913801, 0.62982388760580406825698259195, 1.84291335584595750594774963234, 3.21732584708390013001059052039, 4.56642296726034154094997708058, 6.30844768435841712960388007493, 7.177944783746658203429407118841, 8.22173823874871908778911620322, 8.55491216934592512080968003159, 9.85853784340001995999956926446, 10.85497005884239449731198752435, 12.16829688969151001854363925891, 12.92807292109870940558670845718, 14.457675343628194683569051925728, 15.18238420280862862182905024701, 15.74161868828044602835776015847, 17.11121697749354857776433751085, 17.88056034502013795718484445821, 19.09303422190190389787570799006, 19.383186744532567580273299559, 20.388586334696491502534443903562, 20.98691685205739944530147877007, 23.01483307386979283703818506564, 23.48818115066135356203338149928, 24.566228899239345649521803292299, 25.22878243572032598496727157273

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