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Properties

Label 1-287-287.76-r0-0-0
Degree $1$
Conductor $287$
Sign $0.748 - 0.663i$
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  + (−0.587 − 0.809i)2-s + (−0.707 + 0.707i)3-s + (−0.309 + 0.951i)4-s + (0.951 + 0.309i)5-s + (0.987 + 0.156i)6-s + (0.951 − 0.309i)8-s − i·9-s + (−0.309 − 0.951i)10-s + (−0.891 − 0.453i)11-s + (−0.453 − 0.891i)12-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (0.453 − 0.891i)17-s + (−0.809 + 0.587i)18-s + (0.156 + 0.987i)19-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.707 + 0.707i)3-s + (−0.309 + 0.951i)4-s + (0.951 + 0.309i)5-s + (0.987 + 0.156i)6-s + (0.951 − 0.309i)8-s − i·9-s + (−0.309 − 0.951i)10-s + (−0.891 − 0.453i)11-s + (−0.453 − 0.891i)12-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (0.453 − 0.891i)17-s + (−0.809 + 0.587i)18-s + (0.156 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.748 - 0.663i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.748 - 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7379739599 - 0.2798494726i\)
\(L(\frac12)\) \(\approx\) \(0.7379739599 - 0.2798494726i\)
\(L(1)\) \(\approx\) \(0.7225816855 - 0.1437403898i\)
\(L(1)\) \(\approx\) \(0.7225816855 - 0.1437403898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.453 - 0.891i)T \)
19 \( 1 + (0.156 + 0.987i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.987 + 0.156i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (0.891 - 0.453i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.987 - 0.156i)T \)
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

−25.76376114634730519864720583813, −24.66652689136821891997474027889, −23.91119530027207475791928084947, −23.377805855707487298329582224250, −22.18979198856403063013634419576, −21.247080885738155861349012443338, −19.92565565441532544339894759528, −18.80313037789569526805542914070, −18.24292071984897676342457902868, −17.2524201213625802166073400536, −16.85252071734237062332462265347, −15.772562929640966483277779986949, −14.58930688373539890933424933801, −13.44226844893443044829584714747, −12.96082517841681008212828364359, −11.43230334702947621674546540427, −10.43077594254114479804946941296, −9.49503135166436440937601622982, −8.39964613763360266383057157917, −7.27315135179335287340104949523, −6.46212498125884021528868690728, −5.499492260496457407619292216510, −4.75219388777014199805771320747, −2.20644234989528525823301917331, −1.18782186527017235449712734509, 0.85081406653191313059655811868, 2.56578495457400217739314262203, 3.49903161106789679313965118654, 5.027489537144984320315293733228, 5.85343733599780149115832538041, 7.32487111754180138844069746514, 8.63814402673108838365746138649, 9.71673095574019512062915612098, 10.39361267505848595714294822689, 10.98705813862278468840395046342, 12.19146840757634807184429351424, 13.099890620882833882833067756291, 14.18070805884182480359048015442, 15.563871350249338827750194687616, 16.54710819964833543880553791884, 17.2992232910037696751404624310, 18.22918372225445336173432846156, 18.69379699081851354003405100531, 20.34523192980050448291087456207, 20.97031071386732547323794600542, 21.55375639579738120968299799747, 22.62401744456923089129713090888, 23.07398447873852128919001657608, 24.83172804593556377202375530371, 25.62811128619920915963815771077

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