Properties

Label 2-287-41.32-c1-0-6
Degree $2$
Conductor $287$
Sign $-0.996 + 0.0828i$
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  + 1.43i·2-s + (−1.43 + 1.43i)3-s − 0.0521·4-s + 4.15i·5-s + (−2.04 − 2.04i)6-s + (0.707 − 0.707i)7-s + 2.79i·8-s − 1.09i·9-s − 5.94·10-s + (4.11 − 4.11i)11-s + (0.0746 − 0.0746i)12-s + (0.0880 − 0.0880i)13-s + (1.01 + 1.01i)14-s + (−5.94 − 5.94i)15-s − 4.10·16-s + (−2.13 − 2.13i)17-s + ⋯
L(s)  = 1  + 1.01i·2-s + (−0.825 + 0.825i)3-s − 0.0260·4-s + 1.85i·5-s + (−0.836 − 0.836i)6-s + (0.267 − 0.267i)7-s + 0.986i·8-s − 0.364i·9-s − 1.88·10-s + (1.23 − 1.23i)11-s + (0.0215 − 0.0215i)12-s + (0.0244 − 0.0244i)13-s + (0.270 + 0.270i)14-s + (−1.53 − 1.53i)15-s − 1.02·16-s + (−0.518 − 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.996 + 0.0828i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.996 + 0.0828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0471315 - 1.13611i\)
\(L(\frac12)\) \(\approx\) \(0.0471315 - 1.13611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-5.98 - 2.28i)T \)
good2 \( 1 - 1.43iT - 2T^{2} \)
3 \( 1 + (1.43 - 1.43i)T - 3iT^{2} \)
5 \( 1 - 4.15iT - 5T^{2} \)
11 \( 1 + (-4.11 + 4.11i)T - 11iT^{2} \)
13 \( 1 + (-0.0880 + 0.0880i)T - 13iT^{2} \)
17 \( 1 + (2.13 + 2.13i)T + 17iT^{2} \)
19 \( 1 + (-0.297 - 0.297i)T + 19iT^{2} \)
23 \( 1 - 4.16T + 23T^{2} \)
29 \( 1 + (-5.55 + 5.55i)T - 29iT^{2} \)
31 \( 1 + 0.142T + 31T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
43 \( 1 - 3.03iT - 43T^{2} \)
47 \( 1 + (-3.23 - 3.23i)T + 47iT^{2} \)
53 \( 1 + (2.33 - 2.33i)T - 53iT^{2} \)
59 \( 1 - 3.83T + 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + (-0.401 - 0.401i)T + 67iT^{2} \)
71 \( 1 + (9.25 - 9.25i)T - 71iT^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 + (-4.00 + 4.00i)T - 79iT^{2} \)
83 \( 1 - 6.87T + 83T^{2} \)
89 \( 1 + (10.4 - 10.4i)T - 89iT^{2} \)
97 \( 1 + (5.89 + 5.89i)T + 97iT^{2} \)
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

−11.64807363258470289021725486634, −11.20809663238692979042181363229, −10.74498147492133777211990428569, −9.597675339233132671367449782041, −8.214226212908261128415196265668, −7.06593400070369895262054378914, −6.40714951733508707115010709099, −5.67944697707204987805147083459, −4.23632651675337832072970268213, −2.83691221821338856206742195996, 1.04924076853259506580118537254, 1.81715522153348292320295101128, 4.07324632679035057793918922125, 5.05411707534521169524214743384, 6.35588218593591003774753308173, 7.29133183516281231019953524346, 8.828689823172096693475028140722, 9.379023658359827578585209618291, 10.67901488963885642899148807081, 11.89613333106329586886180808632

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