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Properties

Label 2-287-287.146-c2-0-2
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: $\chi_{287} (146, \cdot )$
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

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

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