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Properties

Label 1-287-287.26-r0-0-0
Degree $1$
Conductor $287$
Sign $0.0206 - 0.999i$
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  + (−0.207 − 0.978i)2-s + (0.258 + 0.965i)3-s + (−0.913 + 0.406i)4-s + (−0.994 − 0.104i)5-s + (0.891 − 0.453i)6-s + (0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.104 + 0.994i)10-s + (−0.777 − 0.629i)11-s + (−0.629 − 0.777i)12-s + (−0.453 − 0.891i)13-s + (−0.156 − 0.987i)15-s + (0.669 − 0.743i)16-s + (0.629 − 0.777i)17-s + (0.669 + 0.743i)18-s + (0.998 − 0.0523i)19-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.258 + 0.965i)3-s + (−0.913 + 0.406i)4-s + (−0.994 − 0.104i)5-s + (0.891 − 0.453i)6-s + (0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.104 + 0.994i)10-s + (−0.777 − 0.629i)11-s + (−0.629 − 0.777i)12-s + (−0.453 − 0.891i)13-s + (−0.156 − 0.987i)15-s + (0.669 − 0.743i)16-s + (0.629 − 0.777i)17-s + (0.669 + 0.743i)18-s + (0.998 − 0.0523i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.0206 - 0.999i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.0206 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5125235121 - 0.5020258813i\)
\(L(\frac12)\) \(\approx\) \(0.5125235121 - 0.5020258813i\)
\(L(1)\) \(\approx\) \(0.7054194782 - 0.2431077017i\)
\(L(1)\) \(\approx\) \(0.7054194782 - 0.2431077017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.994 - 0.104i)T \)
11 \( 1 + (-0.777 - 0.629i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (0.629 - 0.777i)T \)
19 \( 1 + (0.998 - 0.0523i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.987 - 0.156i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.838 - 0.544i)T \)
53 \( 1 + (0.358 - 0.933i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (-0.933 - 0.358i)T \)
71 \( 1 + (0.156 - 0.987i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.0523 - 0.998i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−25.749253271954634104201803248834, −24.74878886737415718598339494891, −23.967224112821546574054476382957, −23.32557530901673702287831645559, −22.749929949908300922620215886696, −21.259857874292791010486256968286, −19.83047656005397496578435595455, −19.24129335104537394559385932130, −18.42755104193807397357886692870, −17.61415364682382541446370808912, −16.55815207753185786712424916487, −15.6026343156129430562376535832, −14.69084217422875420314005880988, −13.96194523170598446053858163398, −12.74539268874626073111844653387, −12.07484388077426133275194110291, −10.66354929362067927223770575715, −9.28634690307737230870283471035, −8.31385803468378974190991992538, −7.40863999222851931121713142588, −6.99038206233794756032119461083, −5.610750615741119280438621215547, −4.43246530709168979976123967146, −3.03657626621397309751139519133, −1.254165168558354757923261241044, 0.58176559562892339218423262510, 2.89011943458620260364560679243, 3.2677922726952027425931295658, 4.64928948319268003231650910247, 5.319795045354461168912893930324, 7.66607713778152118573108013118, 8.28349292958435576351505642026, 9.40325205505120057188831360049, 10.29943660507919666687377274650, 11.167693458535622891103172258632, 11.92837568685008380541097441645, 13.097013483318200295081954750096, 14.1479696724481415270316384878, 15.236124567664653289511705638946, 16.116792019249792157944317794914, 16.99786341865600302761942015242, 18.290294193019017982840071949067, 19.16860907325379289489549308403, 20.05279946978910168289546129898, 20.667911176058012416651756073973, 21.46276828119557079409714195744, 22.62516777309815422325060056846, 22.961846934736153806236209660605, 24.33875681523393488052967335744, 25.661515961949776557147388182500

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