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Properties

Label 2-287-287.230-c2-0-14
Degree $2$
Conductor $287$
Sign $0.0652 - 0.997i$
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  + (−1.07 + 3.31i)2-s − 5.20·3-s + (−6.58 − 4.78i)4-s + (−3.83 + 5.27i)5-s + (5.59 − 17.2i)6-s + (−2.02 − 6.69i)7-s + (11.6 − 8.48i)8-s + 18.0·9-s + (−13.3 − 18.3i)10-s + (−5.80 − 7.98i)11-s + (34.2 + 24.8i)12-s + (−2.67 + 8.24i)13-s + (24.3 + 0.489i)14-s + (19.9 − 27.4i)15-s + (5.47 + 16.8i)16-s + (−19.9 + 14.4i)17-s + ⋯
L(s)  = 1  + (−0.538 + 1.65i)2-s − 1.73·3-s + (−1.64 − 1.19i)4-s + (−0.767 + 1.05i)5-s + (0.933 − 2.87i)6-s + (−0.289 − 0.957i)7-s + (1.45 − 1.06i)8-s + 2.00·9-s + (−1.33 − 1.83i)10-s + (−0.527 − 0.725i)11-s + (2.85 + 2.07i)12-s + (−0.206 + 0.634i)13-s + (1.74 + 0.0349i)14-s + (1.32 − 1.83i)15-s + (0.342 + 1.05i)16-s + (−1.17 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0652 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0652 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.0652 - 0.997i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ 0.0652 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.146788 + 0.137501i\)
\(L(\frac12)\) \(\approx\) \(0.146788 + 0.137501i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.02 + 6.69i)T \)
41 \( 1 + (-40.6 + 5.66i)T \)
good2 \( 1 + (1.07 - 3.31i)T + (-3.23 - 2.35i)T^{2} \)
3 \( 1 + 5.20T + 9T^{2} \)
5 \( 1 + (3.83 - 5.27i)T + (-7.72 - 23.7i)T^{2} \)
11 \( 1 + (5.80 + 7.98i)T + (-37.3 + 115. i)T^{2} \)
13 \( 1 + (2.67 - 8.24i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (19.9 - 14.4i)T + (89.3 - 274. i)T^{2} \)
19 \( 1 + (10.7 + 33.0i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + (7.77 - 23.9i)T + (-427. - 310. i)T^{2} \)
29 \( 1 + (20.1 - 27.7i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-12.9 - 17.7i)T + (-296. + 913. i)T^{2} \)
37 \( 1 + (29.4 + 21.4i)T + (423. + 1.30e3i)T^{2} \)
43 \( 1 + (-2.00 + 6.18i)T + (-1.49e3 - 1.08e3i)T^{2} \)
47 \( 1 + (4.62 - 14.2i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-47.5 + 65.4i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-3.98 - 1.29i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-69.2 + 22.5i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + (19.5 - 26.8i)T + (-1.38e3 - 4.26e3i)T^{2} \)
71 \( 1 + (-1.20 - 1.65i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 - 69.4iT - 5.32e3T^{2} \)
79 \( 1 - 51.3iT - 6.24e3T^{2} \)
83 \( 1 + 3.58iT - 6.88e3T^{2} \)
89 \( 1 + (-6.61 - 20.3i)T + (-6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (-38.1 - 27.7i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.21424477461715940824635939242, −11.04915416145466588390041243418, −10.05033627648820897458494775433, −8.708809853305079913881248809160, −7.27030146780284716015160923868, −6.97713602275760908553589207867, −6.25327969860208291283004009736, −5.13032377039859501225730532873, −4.03916402058278319240129781686, −0.33205758423843590608875625133, 0.52794574881012584439381121562, 2.19078637975476006977893526583, 4.16881697516678543584068084734, 4.90871399431817742760706832302, 6.09419508101522823201726086777, 7.82087388377260899437532468504, 8.835348765765866244055445686323, 9.916511092806667648691392135739, 10.55397446489101888210167680838, 11.63428503194120981700033012082

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