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Properties

Label 2-287-41.18-c1-0-4
Degree $2$
Conductor $287$
Sign $0.929 + 0.369i$
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  + (−0.454 − 1.39i)2-s − 3.02·3-s + (−0.131 + 0.0954i)4-s + (−2.14 + 1.55i)5-s + (1.37 + 4.22i)6-s + (−0.309 + 0.951i)7-s + (−2.18 − 1.58i)8-s + 6.14·9-s + (3.15 + 2.29i)10-s + (3.91 + 2.84i)11-s + (0.397 − 0.288i)12-s + (−1.45 − 4.46i)13-s + 1.47·14-s + (6.48 − 4.71i)15-s + (−1.32 + 4.08i)16-s + (3.40 + 2.47i)17-s + ⋯
L(s)  = 1  + (−0.321 − 0.988i)2-s − 1.74·3-s + (−0.0657 + 0.0477i)4-s + (−0.959 + 0.697i)5-s + (0.561 + 1.72i)6-s + (−0.116 + 0.359i)7-s + (−0.772 − 0.561i)8-s + 2.04·9-s + (0.997 + 0.724i)10-s + (1.17 + 0.856i)11-s + (0.114 − 0.0833i)12-s + (−0.402 − 1.23i)13-s + 0.393·14-s + (1.67 − 1.21i)15-s + (−0.332 + 1.02i)16-s + (0.825 + 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.929 + 0.369i$
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: \((2,\ 287,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503136 - 0.0962557i\)
\(L(\frac12)\) \(\approx\) \(0.503136 - 0.0962557i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.55 - 4.50i)T \)
good2 \( 1 + (0.454 + 1.39i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 3.02T + 3T^{2} \)
5 \( 1 + (2.14 - 1.55i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-3.91 - 2.84i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.45 + 4.46i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.40 - 2.47i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.822 + 2.53i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.859 - 2.64i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.992 + 0.721i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-7.53 - 5.47i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.13 + 5.90i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.41 - 4.33i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.83 - 11.8i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.64 - 2.64i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.03 - 3.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.12 + 6.54i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.72 - 4.88i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.90 + 2.11i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 7.93T + 73T^{2} \)
79 \( 1 + 8.47T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 + (0.230 - 0.709i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.74 + 4.90i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.63953733671162640850603787152, −11.02447852800742654043302568057, −10.24870609279177717209058636739, −9.504062781884124680614296073287, −7.67317366837119673371786328550, −6.69037844662013678000943060801, −5.86774894815721812706004697357, −4.47467617552934053611768264898, −3.12678549898068091190352980181, −1.07603992771607723704592648919, 0.69779854688895481131756611893, 3.98166877624149470844669140097, 5.01132630574993001332181400234, 6.17160449397054739716562258586, 6.77645174111296460876135451846, 7.74386442149768568844911003048, 8.824111215266625363625203726506, 9.973532285675011202594866638693, 11.37030195877688720296281137992, 11.90623754517319206513050859541

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