Properties

Label 1-287-287.67-r1-0-0
Degree $1$
Conductor $287$
Sign $0.996 + 0.0839i$
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  + (−0.743 + 0.669i)2-s + (0.965 + 0.258i)3-s + (0.104 − 0.994i)4-s + (−0.406 − 0.913i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.913 + 0.406i)10-s + (0.933 − 0.358i)11-s + (0.358 − 0.933i)12-s + (0.453 + 0.891i)13-s + (−0.156 − 0.987i)15-s + (−0.978 − 0.207i)16-s + (−0.358 − 0.933i)17-s + (−0.978 + 0.207i)18-s + (0.544 + 0.838i)19-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.965 + 0.258i)3-s + (0.104 − 0.994i)4-s + (−0.406 − 0.913i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.913 + 0.406i)10-s + (0.933 − 0.358i)11-s + (0.358 − 0.933i)12-s + (0.453 + 0.891i)13-s + (−0.156 − 0.987i)15-s + (−0.978 − 0.207i)16-s + (−0.358 − 0.933i)17-s + (−0.978 + 0.207i)18-s + (0.544 + 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.996 + 0.0839i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.996 + 0.0839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.936797423 + 0.08145994171i\)
\(L(\frac12)\) \(\approx\) \(1.936797423 + 0.08145994171i\)
\(L(1)\) \(\approx\) \(1.109723534 + 0.1593189164i\)
\(L(1)\) \(\approx\) \(1.109723534 + 0.1593189164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-0.406 - 0.913i)T \)
11 \( 1 + (0.933 - 0.358i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
17 \( 1 + (-0.358 - 0.933i)T \)
19 \( 1 + (0.544 + 0.838i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.987 - 0.156i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (0.0523 - 0.998i)T \)
53 \( 1 + (0.629 + 0.777i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.207 - 0.978i)T \)
67 \( 1 + (0.777 - 0.629i)T \)
71 \( 1 + (0.156 - 0.987i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (0.838 - 0.544i)T \)
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

−25.7414600043214065348422731692, −24.72809551700157781662576341235, −23.49640317651867477200411096204, −22.221411265952683495388313413, −21.65430006909706422572770766172, −20.324259154445908317848955705390, −19.760241861434920856512993972824, −19.17664946560020435785605373767, −18.04014023346639351990902359281, −17.61785889732676265791144981233, −15.9642299839213501803634377013, −15.18524557617085038086700653685, −14.13739452077933910755578995850, −13.103004919574699047757667518652, −12.12338881590480449286784074000, −11.08492769088482726162128084175, −10.13809242872079181773737888989, −9.18528389383939817557193475970, −8.19591010751536946705474394301, −7.3842328846689140566086346912, −6.45890739201787453989996441184, −4.10042515739882440366392659162, −3.34632473160947415930189091013, −2.34253803625794034620800560754, −1.08231336871330124850413892227, 0.84455368318739879645431923616, 2.02451324688887499424542801963, 3.846523651153645692780928852123, 4.7607773083092600890947628502, 6.19660449224363319810538430671, 7.406082700073462992904429146706, 8.30801112735572518977272178917, 9.09379879981961670897696005896, 9.64286750382885375870564826724, 11.070521086304105275617416278353, 12.19897672730591591735301235870, 13.73149586443555751109199115861, 14.23018128744845115279129548363, 15.37516732946755651610887225353, 16.34345123670999410095029598406, 16.59516984696848656311160509836, 18.15523935703351312728929469830, 18.96494163114303609137298798475, 19.92250197249453715794963016329, 20.36105206495731296963735406034, 21.48421345286467044953782943716, 22.82753456708646285988576938312, 23.965899196353615055259456758612, 24.68799471494244506848717206784, 25.17651013615362460822791034858

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