↑

Properties

Label 2-588-196.55-c1-0-17
Degree $2$
Conductor $588$
Sign $0.961 - 0.273i$
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  + (0.761 − 1.19i)2-s + (0.900 + 0.433i)3-s + (−0.839 − 1.81i)4-s + (−1.00 + 2.07i)5-s + (1.20 − 0.743i)6-s + (−0.0377 + 2.64i)7-s + (−2.80 − 0.381i)8-s + (0.623 + 0.781i)9-s + (1.71 + 2.77i)10-s + (4.29 + 3.42i)11-s + (0.0308 − 1.99i)12-s + (2.35 + 1.87i)13-s + (3.12 + 2.05i)14-s + (−1.80 + 1.43i)15-s + (−2.58 + 3.04i)16-s + (−6.03 − 1.37i)17-s + ⋯
L(s)  = 1  + (0.538 − 0.842i)2-s + (0.520 + 0.250i)3-s + (−0.419 − 0.907i)4-s + (−0.447 + 0.928i)5-s + (0.491 − 0.303i)6-s + (−0.0142 + 0.999i)7-s + (−0.990 − 0.134i)8-s + (0.207 + 0.260i)9-s + (0.541 + 0.877i)10-s + (1.29 + 1.03i)11-s + (0.00890 − 0.577i)12-s + (0.653 + 0.521i)13-s + (0.834 + 0.550i)14-s + (−0.465 + 0.371i)15-s + (−0.647 + 0.762i)16-s + (−1.46 − 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.961 - 0.273i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99658 + 0.277900i\)
\(L(\frac12)\) \(\approx\) \(1.99658 + 0.277900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.761 + 1.19i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (0.0377 - 2.64i)T \)
good5 \( 1 + (1.00 - 2.07i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-4.29 - 3.42i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.35 - 1.87i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (6.03 + 1.37i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 0.661T + 19T^{2} \)
23 \( 1 + (-2.93 + 0.670i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.91 + 8.36i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 6.10T + 31T^{2} \)
37 \( 1 + (-0.683 + 2.99i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (1.63 - 3.38i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.84 + 3.83i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (3.94 - 4.94i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.23 + 5.39i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (11.5 - 5.57i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-13.5 - 3.10i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 4.63iT - 67T^{2} \)
71 \( 1 + (-8.42 + 1.92i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-6.79 + 5.41i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 7.80iT - 79T^{2} \)
83 \( 1 + (4.75 + 5.96i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.28 - 1.02i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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

−11.01344607477397175783000967697, −9.809320340728720767362044348005, −9.231039538012037762886117699961, −8.417866159135281033544461976029, −6.83370816532261952051191058690, −6.30274931199832462917217820026, −4.71139273998391449169767597896, −4.00024082445731792208805550460, −2.85808232069175406599851771244, −1.94374966399373792682451635856, 0.988922915586996984301746840791, 3.30740246734624980406402932541, 4.06711998214866636563764358603, 4.93693979060738797984271275094, 6.40178416916873453357565764920, 6.87221958841234435138057244131, 8.224098138766854006871511255908, 8.515809750765567938097876076281, 9.309590943393198458533130077140, 10.82291452701058090898635375729

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