↑

Properties

Label 2-588-28.19-c1-0-38
Degree $2$
Conductor $588$
Sign $-0.958 - 0.284i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (1.15 − 0.809i)2-s + (−0.5 − 0.866i)3-s + (0.689 − 1.87i)4-s + (−2.88 − 1.66i)5-s + (−1.28 − 0.599i)6-s + (−0.719 − 2.73i)8-s + (−0.499 + 0.866i)9-s + (−4.69 + 0.403i)10-s + (−0.810 + 0.468i)11-s + (−1.97 + 0.341i)12-s + 1.87i·13-s + 3.33i·15-s + (−3.04 − 2.59i)16-s + (−4.51 + 2.60i)17-s + (0.121 + 1.40i)18-s + (3.56 − 6.16i)19-s + ⋯
L(s)  = 1  + (0.820 − 0.572i)2-s + (−0.288 − 0.499i)3-s + (0.344 − 0.938i)4-s + (−1.29 − 0.745i)5-s + (−0.522 − 0.244i)6-s + (−0.254 − 0.967i)8-s + (−0.166 + 0.288i)9-s + (−1.48 + 0.127i)10-s + (−0.244 + 0.141i)11-s + (−0.568 + 0.0984i)12-s + 0.519i·13-s + 0.861i·15-s + (−0.762 − 0.647i)16-s + (−1.09 + 0.631i)17-s + (0.0285 + 0.332i)18-s + (0.817 − 1.41i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.958 - 0.284i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.958 - 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156595 + 1.07683i\)
\(L(\frac12)\) \(\approx\) \(0.156595 + 1.07683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.809i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (2.88 + 1.66i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.810 - 0.468i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.87iT - 13T^{2} \)
17 \( 1 + (4.51 - 2.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 + 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.810 + 0.468i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.561 - 0.972i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 + 9.06iT - 43T^{2} \)
47 \( 1 + (-3.12 + 5.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.12 + 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.15 + 2.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.47 + 5.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.86iT - 71T^{2} \)
73 \( 1 + (-5.77 + 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.07 + 1.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (1.26 + 0.731i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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

−10.65460648078313362249786031551, −9.334038874053680999700101149000, −8.475838255553107575463442027955, −7.32689993891291126206379873648, −6.60130014874693254910544223732, −5.26294412549321973490011058552, −4.53768014685917566209087897428, −3.58920244086481497338365569587, −2.08971545945485116531926123036, −0.46982035789282663724647482727, 2.86840158105186270366427343072, 3.70767645992710195760021247551, 4.56570203540170260444098438825, 5.65092743703466155395385041483, 6.61949753756518947800574647188, 7.61326094916271106688482910106, 8.089024875146096646526448382090, 9.380089512036879628727038744025, 10.67581825813464557589410485795, 11.26804311081734977498994148959

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