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Properties

Label 2-287-41.16-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: $\chi_{287} (57, \cdot )$
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

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

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