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Properties

Label 2-287-41.3-c2-0-18
Degree $2$
Conductor $287$
Sign $0.909 - 0.416i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 + 1.56i)2-s + (−1.33 + 3.23i)3-s − 0.878i·4-s + (−2.93 − 2.93i)5-s + (−2.95 − 7.13i)6-s + (1.01 − 2.44i)7-s + (−4.87 − 4.87i)8-s + (−2.28 − 2.28i)9-s + 9.17·10-s + (4.46 + 1.84i)11-s + (2.83 + 1.17i)12-s + (6.92 − 16.7i)13-s + (2.23 + 5.39i)14-s + (13.4 − 5.56i)15-s + 18.7·16-s + (2.72 + 6.57i)17-s + ⋯
L(s)  = 1  + (−0.780 + 0.780i)2-s + (−0.446 + 1.07i)3-s − 0.219i·4-s + (−0.587 − 0.587i)5-s + (−0.492 − 1.18i)6-s + (0.144 − 0.349i)7-s + (−0.609 − 0.609i)8-s + (−0.254 − 0.254i)9-s + 0.917·10-s + (0.406 + 0.168i)11-s + (0.236 + 0.0979i)12-s + (0.532 − 1.28i)13-s + (0.159 + 0.385i)14-s + (0.895 − 0.370i)15-s + 1.17·16-s + (0.160 + 0.386i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ 0.909 - 0.416i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.630967 + 0.137596i\)
\(L(\frac12)\) \(\approx\) \(0.630967 + 0.137596i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.01 + 2.44i)T \)
41 \( 1 + (-40.7 + 4.57i)T \)
good2 \( 1 + (1.56 - 1.56i)T - 4iT^{2} \)
3 \( 1 + (1.33 - 3.23i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (2.93 + 2.93i)T + 25iT^{2} \)
11 \( 1 + (-4.46 - 1.84i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (-6.92 + 16.7i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + (-2.72 - 6.57i)T + (-204. + 204. i)T^{2} \)
19 \( 1 + (12.9 + 31.1i)T + (-255. + 255. i)T^{2} \)
23 \( 1 - 24.5iT - 529T^{2} \)
29 \( 1 + (-4.84 + 11.6i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 28.5iT - 961T^{2} \)
37 \( 1 - 9.64T + 1.36e3T^{2} \)
43 \( 1 + (-7.52 + 7.52i)T - 1.84e3iT^{2} \)
47 \( 1 + (19.8 + 47.8i)T + (-1.56e3 + 1.56e3i)T^{2} \)
53 \( 1 + (-0.988 - 0.409i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 - 104.T + 3.48e3T^{2} \)
61 \( 1 + (-27.1 + 27.1i)T - 3.72e3iT^{2} \)
67 \( 1 + (44.6 + 107. i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (31.9 - 77.1i)T + (-3.56e3 - 3.56e3i)T^{2} \)
73 \( 1 + (-78.5 + 78.5i)T - 5.32e3iT^{2} \)
79 \( 1 + (-13.8 - 5.74i)T + (4.41e3 + 4.41e3i)T^{2} \)
83 \( 1 + 33.3T + 6.88e3T^{2} \)
89 \( 1 + (-9.30 + 22.4i)T + (-5.60e3 - 5.60e3i)T^{2} \)
97 \( 1 + (-42.8 + 17.7i)T + (6.65e3 - 6.65e3i)T^{2} \)
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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

−11.39347158596841164553977125416, −10.53975678182076607067838284991, −9.642654878599893854868307526145, −8.720214028865342665526425657362, −7.974922567887399901214919370162, −6.93121022284092994212692459881, −5.64541123432091117490019843607, −4.52000906736199706621342869550, −3.52103212345050108195097522051, −0.52911943416210363184734384598, 1.15984432952064586673754641552, 2.30248264822906143352531053661, 3.95198310813745292502441727230, 5.89598106859437421378900530855, 6.62715355946259164787226839498, 7.77273322630735548258548743076, 8.707744038834190906898459532898, 9.719231873695147081489801594309, 10.85095867021227445129809746280, 11.55715008525139722153065582141

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