↑

Properties

Label 2-287-7.4-c1-0-22
Degree $2$
Conductor $287$
Sign $-0.957 + 0.289i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.678 − 1.17i)2-s + (1.11 − 1.92i)3-s + (0.0800 − 0.138i)4-s + (−1.11 − 1.92i)5-s − 3.02·6-s + (2.60 + 0.437i)7-s − 2.92·8-s + (−0.981 − 1.70i)9-s + (−1.51 + 2.61i)10-s + (0.564 − 0.977i)11-s + (−0.178 − 0.308i)12-s + 2.22·13-s + (−1.25 − 3.36i)14-s − 4.96·15-s + (1.82 + 3.16i)16-s + (−2.02 + 3.51i)17-s + ⋯
L(s)  = 1  + (−0.479 − 0.830i)2-s + (0.643 − 1.11i)3-s + (0.0400 − 0.0692i)4-s + (−0.498 − 0.862i)5-s − 1.23·6-s + (0.986 + 0.165i)7-s − 1.03·8-s + (−0.327 − 0.566i)9-s + (−0.477 + 0.827i)10-s + (0.170 − 0.294i)11-s + (−0.0514 − 0.0891i)12-s + 0.617·13-s + (−0.335 − 0.898i)14-s − 1.28·15-s + (0.456 + 0.791i)16-s + (−0.491 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185742 - 1.25460i\)
\(L(\frac12)\) \(\approx\) \(0.185742 - 1.25460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.60 - 0.437i)T \)
41 \( 1 + T \)
good2 \( 1 + (0.678 + 1.17i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.11 + 1.92i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.11 + 1.92i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.564 + 0.977i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 + (2.02 - 3.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.80 - 3.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.34 - 2.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
31 \( 1 + (0.172 - 0.299i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.67 + 2.90i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + (-3.65 - 6.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.17 + 7.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.61 + 4.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.80 + 4.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.41 + 7.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.92T + 71T^{2} \)
73 \( 1 + (0.407 - 0.706i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.42 - 7.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 + (-7.88 - 13.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.10T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47479671193510480280017914830, −10.67224987593786034254634546742, −9.290974021902520640173100524404, −8.400336053956682845904963042167, −8.027661047314228206857440193148, −6.60019304522052790335773064074, −5.30391237679442491310631547714, −3.65820300316052625142935505530, −2.01553587732130678935907064399, −1.18760131452415854738669706936, 2.83305577190805113889428931608, 3.88448479431798720993026874451, 5.10080870315216375890931933431, 6.75623236107572974200793179763, 7.42758155086179532754902343639, 8.514020884142501271913062939796, 9.091816622472562791868659249792, 10.23507117813507083181903811376, 11.20657267513683344248015043039, 11.83905430778535620895465396541

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