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Properties

Label 1-287-287.54-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.398 + 0.917i$
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  + (0.406 − 0.913i)2-s + (−0.965 + 0.258i)3-s + (−0.669 − 0.743i)4-s + (0.207 + 0.978i)5-s + (−0.156 + 0.987i)6-s + (−0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.978 + 0.207i)10-s + (−0.544 − 0.838i)11-s + (0.838 + 0.544i)12-s + (−0.987 − 0.156i)13-s + (−0.453 − 0.891i)15-s + (−0.104 + 0.994i)16-s + (−0.838 + 0.544i)17-s + (−0.104 − 0.994i)18-s + (0.629 + 0.777i)19-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.965 + 0.258i)3-s + (−0.669 − 0.743i)4-s + (0.207 + 0.978i)5-s + (−0.156 + 0.987i)6-s + (−0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.978 + 0.207i)10-s + (−0.544 − 0.838i)11-s + (0.838 + 0.544i)12-s + (−0.987 − 0.156i)13-s + (−0.453 − 0.891i)15-s + (−0.104 + 0.994i)16-s + (−0.838 + 0.544i)17-s + (−0.104 − 0.994i)18-s + (0.629 + 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.398 + 0.917i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.398 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09847435648 + 0.1501506255i\)
\(L(\frac12)\) \(\approx\) \(0.09847435648 + 0.1501506255i\)
\(L(1)\) \(\approx\) \(0.6150239681 - 0.1484272405i\)
\(L(1)\) \(\approx\) \(0.6150239681 - 0.1484272405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (-0.544 - 0.838i)T \)
13 \( 1 + (-0.987 - 0.156i)T \)
17 \( 1 + (-0.838 + 0.544i)T \)
19 \( 1 + (0.629 + 0.777i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (-0.891 + 0.453i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.933 - 0.358i)T \)
53 \( 1 + (-0.0523 + 0.998i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.994 - 0.104i)T \)
67 \( 1 + (-0.998 - 0.0523i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.777 + 0.629i)T \)
97 \( 1 + (0.453 + 0.891i)T \)
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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

−24.92375329908583180333736905780, −24.1403236534181280358630069124, −23.79062105992099939037326541862, −22.54640793451820609473819810228, −22.04839075030707507832830357483, −20.969699373106564111674665690666, −19.89676522101293384304613899007, −18.38768954121331842328430071434, −17.56404002639856642572195533237, −17.09115729377051057138668136631, −15.99008882314284593415174505561, −15.498336153644162049962743811032, −13.99964499158767864696584357783, −13.0650068093967498956645498672, −12.42207306854932372798702114590, −11.556870893601970825940827811885, −9.92577467373206944380685161217, −9.06837806997727577631860432235, −7.61413641830771524958040212445, −7.030481704861672169034433762288, −5.64987602402608309498143599598, −5.03256916777279089289365747140, −4.21787588087385692819165200418, −2.11299208503215986228595894176, −0.11120730694318085019533776774, 1.843431739937915381920719560993, 3.1296235306410966134633641443, 4.20303764838872471120404540890, 5.49594904398746946409187161138, 6.12218485586937413825921268660, 7.487630897784085476092121760898, 9.20859538346958011858683935765, 10.33038327325544819299028508213, 10.719800478036102529182764840831, 11.68976728385755743156633651607, 12.56814166221876253197198606824, 13.64650038195339446892233396395, 14.642206571895657208927905055391, 15.52928955548305363384617319516, 16.78531571956426433874682391700, 17.951885768414169921667585845101, 18.44805981739483430873177264109, 19.4058513490964405644416015656, 20.58151125412951871961190980062, 21.64029416561506034588588835472, 22.1506745100465692085499706047, 22.71072383571322950589514880344, 23.83076624392277641756735539664, 24.43788171601210028128441608748, 26.273854554659836544813142597657

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