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Properties

Label 1-287-287.153-r0-0-0
Degree $1$
Conductor $287$
Sign $0.930 + 0.367i$
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  + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.891 + 0.453i)6-s + (0.587 − 0.809i)8-s + i·9-s + (0.809 + 0.587i)10-s + (−0.156 − 0.987i)11-s + (0.987 + 0.156i)12-s + (−0.453 + 0.891i)13-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + (−0.987 + 0.156i)17-s + (0.309 + 0.951i)18-s + (−0.453 − 0.891i)19-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.891 + 0.453i)6-s + (0.587 − 0.809i)8-s + i·9-s + (0.809 + 0.587i)10-s + (−0.156 − 0.987i)11-s + (0.987 + 0.156i)12-s + (−0.453 + 0.891i)13-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + (−0.987 + 0.156i)17-s + (0.309 + 0.951i)18-s + (−0.453 − 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.930 + 0.367i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.930 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.808982189 + 0.5346249633i\)
\(L(\frac12)\) \(\approx\) \(2.808982189 + 0.5346249633i\)
\(L(1)\) \(\approx\) \(2.244556491 + 0.2373587212i\)
\(L(1)\) \(\approx\) \(2.244556491 + 0.2373587212i\)

Euler product

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

−25.25084553926725365637985308777, −24.67610181789919229702970321119, −23.78089387221295268513855966001, −22.98057333655721127160641282956, −21.848508851869820626778453075523, −20.84305542387215423820122263239, −20.24239217169375196922833334903, −19.5211788328200932963484169473, −17.776575409478262263685780018380, −17.45963651132905665106577874788, −16.01548432752756216294691055140, −15.15500841053110587879137046498, −14.218500120573239118945575655059, −13.38120097767135907608038853189, −12.60005440239335857172618379967, −12.11651299109305792138926169865, −10.41259458588837563361809222426, −9.16033432004688718026702961081, −8.078258214258828126855818297328, −7.22856864529009273864113486321, −6.11153215977285298710516858062, −5.07611383358597690370109669470, −3.93321015865481291926491111153, −2.53314757622130762459349025497, −1.68199789060589796780203946340, 2.122581915529336730433297515067, 2.78100512143485893750702986036, 3.93890924472879166374978282633, 4.91567548389942810608154726628, 6.18024143078707977308068845907, 7.08390174775957743761743381531, 8.65653123536913521645079456617, 9.726191004148491315072115611626, 10.75435803185501906338670306056, 11.25289610026435095996173684653, 12.82966276238779580414269979747, 13.89264289502548426638675574186, 14.20799728669809277329521326755, 15.24751625607123947098300597391, 16.01604196956344691791419949375, 17.15049680784683302076311993162, 18.74505878069502908969179157458, 19.3589911098035022821906850327, 20.37530367360711652324456401296, 21.339430586524818093575994033864, 21.92483695166892876877012492399, 22.39075360378401365008442478389, 23.85348894753564653082270520581, 24.55005322458230629922400491323, 25.6571296747264456780036575823

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