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Properties

Label 2-287-41.18-c1-0-8
Degree $2$
Conductor $287$
Sign $-0.725 - 0.687i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 2.48i)2-s + 2.61·3-s + (−3.92 + 2.85i)4-s + (−0.5 + 0.363i)5-s + (2.11 + 6.51i)6-s + (0.309 − 0.951i)7-s + (−6.04 − 4.39i)8-s + 3.85·9-s + (−1.30 − 0.951i)10-s + (2.30 + 1.67i)11-s + (−10.2 + 7.46i)12-s + (−0.690 − 2.12i)13-s + 2.61·14-s + (−1.30 + 0.951i)15-s + (3.04 − 9.37i)16-s + (−6.04 − 4.39i)17-s + ⋯
L(s)  = 1  + (0.572 + 1.76i)2-s + 1.51·3-s + (−1.96 + 1.42i)4-s + (−0.223 + 0.162i)5-s + (0.864 + 2.66i)6-s + (0.116 − 0.359i)7-s + (−2.13 − 1.55i)8-s + 1.28·9-s + (−0.413 − 0.300i)10-s + (0.696 + 0.505i)11-s + (−2.96 + 2.15i)12-s + (−0.191 − 0.589i)13-s + 0.699·14-s + (−0.337 + 0.245i)15-s + (0.761 − 2.34i)16-s + (−1.46 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.725 - 0.687i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.725 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842403 + 2.11320i\)
\(L(\frac12)\) \(\approx\) \(0.842403 + 2.11320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (5.89 - 2.48i)T \)
good2 \( 1 + (-0.809 - 2.48i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 + (0.5 - 0.363i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-2.30 - 1.67i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.690 + 2.12i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.04 + 4.39i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.61 + 4.97i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.30 - 4.02i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.23 + 2.35i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.80 - 2.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.16 + 4.47i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (1.38 + 4.25i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.64 - 8.14i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.47 - 3.97i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.35 + 7.24i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.54 + 10.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (11.2 - 8.19i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-12.7 - 9.28i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 0.909T + 79T^{2} \)
83 \( 1 + 0.145T + 83T^{2} \)
89 \( 1 + (-1.88 + 5.79i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.97 - 4.33i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.85343977396777456859303695304, −11.41193077365080707935948331614, −9.522526133185033457215317986956, −9.066313859955594835724717480164, −8.030414176487168579927566510212, −7.31419191353050114175718362758, −6.66740374163055553758103360411, −5.00516544330539303012457879420, −4.10934945992863810767101232395, −2.93010066354813098177444970277, 1.70968128980675454571004916095, 2.71217726371175117090796468275, 3.83005199256922868942562802807, 4.52356652463669628341558211124, 6.30455207848498490284342400318, 8.302340021519677497257497668438, 8.763273131669046000877233040042, 9.665025207669676813202286580922, 10.55694907877723818295818382112, 11.66286446947056043609953275617

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