↑

Properties

Label 2-287-41.37-c1-0-1
Degree $2$
Conductor $287$
Sign $-0.999 - 0.0241i$
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  + (−1.91 + 1.38i)2-s − 1.05·3-s + (1.10 − 3.40i)4-s + (0.886 − 2.72i)5-s + (2.01 − 1.46i)6-s + (−0.809 − 0.587i)7-s + (1.15 + 3.55i)8-s − 1.88·9-s + (2.09 + 6.44i)10-s + (0.763 + 2.34i)11-s + (−1.16 + 3.59i)12-s + (−3.82 + 2.77i)13-s + 2.36·14-s + (−0.935 + 2.87i)15-s + (−1.34 − 0.978i)16-s + (2.33 + 7.19i)17-s + ⋯
L(s)  = 1  + (−1.35 + 0.981i)2-s − 0.609·3-s + (0.553 − 1.70i)4-s + (0.396 − 1.22i)5-s + (0.823 − 0.598i)6-s + (−0.305 − 0.222i)7-s + (0.408 + 1.25i)8-s − 0.628·9-s + (0.662 + 2.03i)10-s + (0.230 + 0.708i)11-s + (−0.337 + 1.03i)12-s + (−1.06 + 0.770i)13-s + 0.631·14-s + (−0.241 + 0.743i)15-s + (−0.336 − 0.244i)16-s + (0.567 + 1.74i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.999 - 0.0241i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.999 - 0.0241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00170506 + 0.141129i\)
\(L(\frac12)\) \(\approx\) \(0.00170506 + 0.141129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-5.15 + 3.80i)T \)
good2 \( 1 + (1.91 - 1.38i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + 1.05T + 3T^{2} \)
5 \( 1 + (-0.886 + 2.72i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (-0.763 - 2.34i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.82 - 2.77i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.33 - 7.19i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.03 + 3.66i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.51 - 1.10i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.53 - 7.79i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.75 + 5.40i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.185 + 0.571i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (6.36 - 4.62i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (3.09 - 2.25i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.56 - 10.9i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.126 - 0.0921i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.358 - 0.260i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.807 + 2.48i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.71 + 5.26i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 8.41T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 5.57T + 83T^{2} \)
89 \( 1 + (9.53 + 6.92i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.326 - 1.00i)T + (-78.4 - 57.0i)T^{2} \)
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

−12.35301243955588167167287009844, −11.01070969553408701140099115120, −10.06632364523842170345536262388, −9.208881998881720849231490326726, −8.637948416924936046873935483358, −7.52354331761527421344554200354, −6.46938381401929074868787862466, −5.66377502591890930803961793185, −4.50914595935309518896640594887, −1.63635076014751102024985017038, 0.16974281404522101462670664392, 2.44814414281637482580817750076, 3.19080492457948550244298166521, 5.48039504345678308496172196087, 6.56809627499405813027392140887, 7.68452938336948675383369597885, 8.701806531075695954009140363697, 9.929013800980224911795547045294, 10.22613477405458195575731860812, 11.29382941494115729711493809953

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