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Properties

Label 2-287-41.18-c1-0-3
Degree $2$
Conductor $287$
Sign $-0.769 + 0.638i$
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  + (0.728 + 2.24i)2-s − 0.551·3-s + (−2.88 + 2.09i)4-s + (−1.91 + 1.38i)5-s + (−0.401 − 1.23i)6-s + (0.309 − 0.951i)7-s + (−2.98 − 2.16i)8-s − 2.69·9-s + (−4.51 − 3.27i)10-s + (−3.63 − 2.64i)11-s + (1.58 − 1.15i)12-s + (1.63 + 5.02i)13-s + 2.35·14-s + (1.05 − 0.766i)15-s + (0.486 − 1.49i)16-s + (4.58 + 3.33i)17-s + ⋯
L(s)  = 1  + (0.515 + 1.58i)2-s − 0.318·3-s + (−1.44 + 1.04i)4-s + (−0.855 + 0.621i)5-s + (−0.164 − 0.504i)6-s + (0.116 − 0.359i)7-s + (−1.05 − 0.766i)8-s − 0.898·9-s + (−1.42 − 1.03i)10-s + (−1.09 − 0.796i)11-s + (0.458 − 0.333i)12-s + (0.452 + 1.39i)13-s + 0.630·14-s + (0.272 − 0.197i)15-s + (0.121 − 0.374i)16-s + (1.11 + 0.808i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.769 + 0.638i$
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: \((2,\ 287,\ (\ :1/2),\ -0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285247 - 0.790055i\)
\(L(\frac12)\) \(\approx\) \(0.285247 - 0.790055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 - 6.15i)T \)
good2 \( 1 + (-0.728 - 2.24i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 0.551T + 3T^{2} \)
5 \( 1 + (1.91 - 1.38i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (3.63 + 2.64i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.63 - 5.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.58 - 3.33i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0994 - 0.305i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.0339 + 0.104i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.41 - 1.02i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.47 + 2.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.75 + 4.90i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.95 - 6.02i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.58 - 7.96i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.54 - 6.20i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.60 - 4.93i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.430 + 1.32i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.73 + 3.44i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (12.2 + 8.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 6.93T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 7.59T + 83T^{2} \)
89 \( 1 + (-3.87 + 11.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.02 + 3.64i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.61756282262138270108176985826, −11.31714910632458267989768640087, −10.86637625481513202707487004730, −9.136510414310037404600747565445, −7.965869047040269092284821275645, −7.63338166374515905525536235132, −6.32643904263447642588085230726, −5.72650499783633839729675157311, −4.42801527705032288124591822322, −3.33333446640431937498622454201, 0.55427298178733024346450246297, 2.55980989613272266008029186407, 3.59157551889704836258814502195, 5.01200077545094685550220428337, 5.48308931715338440034033280223, 7.64112876243841304601136859686, 8.503987882728803131372024985391, 9.757683153376136513835868367446, 10.61148246532531640454621048611, 11.40837396803186339149198923912

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