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Properties

Label 1-287-287.158-r1-0-0
Degree $1$
Conductor $287$
Sign $0.965 - 0.261i$
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  + (0.994 − 0.104i)2-s + (−0.965 − 0.258i)3-s + (0.978 − 0.207i)4-s + (0.743 − 0.669i)5-s + (−0.987 − 0.156i)6-s + (0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (0.669 − 0.743i)10-s + (0.0523 + 0.998i)11-s + (−0.998 − 0.0523i)12-s + (−0.156 + 0.987i)13-s + (−0.891 + 0.453i)15-s + (0.913 − 0.406i)16-s + (0.998 − 0.0523i)17-s + (0.913 + 0.406i)18-s + (0.933 + 0.358i)19-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.965 − 0.258i)3-s + (0.978 − 0.207i)4-s + (0.743 − 0.669i)5-s + (−0.987 − 0.156i)6-s + (0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (0.669 − 0.743i)10-s + (0.0523 + 0.998i)11-s + (−0.998 − 0.0523i)12-s + (−0.156 + 0.987i)13-s + (−0.891 + 0.453i)15-s + (0.913 − 0.406i)16-s + (0.998 − 0.0523i)17-s + (0.913 + 0.406i)18-s + (0.933 + 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.965 - 0.261i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.487184886 - 0.4642339570i\)
\(L(\frac12)\) \(\approx\) \(3.487184886 - 0.4642339570i\)
\(L(1)\) \(\approx\) \(1.866353940 - 0.2522909597i\)
\(L(1)\) \(\approx\) \(1.866353940 - 0.2522909597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.743 - 0.669i)T \)
11 \( 1 + (0.0523 + 0.998i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (0.998 - 0.0523i)T \)
19 \( 1 + (0.933 + 0.358i)T \)
23 \( 1 + (0.104 + 0.994i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.629 - 0.777i)T \)
53 \( 1 + (0.544 - 0.838i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.406 + 0.913i)T \)
67 \( 1 + (-0.838 - 0.544i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (0.358 - 0.933i)T \)
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

−25.067448977888722488320333878350, −24.312546469821819010106180559830, −23.34694908823980949220658518640, −22.4488872740899801330526728398, −22.01523928336636672814396012325, −21.20937488450148027867033510651, −20.28158302609188111500240175368, −18.80898191842926870305326504388, −17.93791732195983166457479102764, −16.83454383592835663303363873855, −16.214292343263584061592311393772, −15.06220127862243048574493628287, −14.27960868270088096311298624510, −13.20635499354842162028008084730, −12.384800465324721542160507325161, −11.18888587408332229972543674462, −10.682645036641807681295176919075, −9.61387773198793943049817798081, −7.73875208751950295296862650674, −6.68469392094673312155529870058, −5.71691141510855830457558571745, −5.27667016487777125873392238834, −3.71562387989230136527798980374, −2.74398627953740946111670888460, −1.03826110002066190423141623010, 1.23508964885610055899993472484, 2.066004047322689930385914154807, 3.91585627203627784250042116403, 5.04654946030532628125219483298, 5.59237654224916486039094000754, 6.729470637965320919692494821957, 7.60427981466093681850388929017, 9.56337408829599198507651470933, 10.25865879624134917605056500686, 11.72555474585254730061622757516, 12.09656318815123380885524985257, 13.14287229749098824213337431697, 13.86633107606973564056125197674, 15.03625390742005229212822686182, 16.27093341073803087633698752641, 16.789263533687839494554105344337, 17.785829746872032687117686654537, 18.90755422515032888119347050914, 20.11599885946241691523022731696, 21.05012168801717781803141272227, 21.71259237146051152665459825884, 22.59946851344576868550794431151, 23.47072802009154728800659942345, 24.10954639019518917830249513309, 25.05208071182974526808498030603

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