↑

Properties

Label 2-588-196.27-c1-0-17
Degree $2$
Conductor $588$
Sign $-0.810 - 0.585i$
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  + (0.981 + 1.01i)2-s + (−0.222 + 0.974i)3-s + (−0.0741 + 1.99i)4-s + (2.53 + 0.578i)5-s + (−1.21 + 0.730i)6-s + (−1.72 + 2.00i)7-s + (−2.10 + 1.88i)8-s + (−0.900 − 0.433i)9-s + (1.89 + 3.14i)10-s + (−0.428 − 0.889i)11-s + (−1.93 − 0.517i)12-s + (0.697 + 1.44i)13-s + (−3.73 + 0.209i)14-s + (−1.12 + 2.34i)15-s + (−3.98 − 0.296i)16-s + (3.94 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.693 + 0.720i)2-s + (−0.128 + 0.562i)3-s + (−0.0370 + 0.999i)4-s + (1.13 + 0.258i)5-s + (−0.494 + 0.298i)6-s + (−0.652 + 0.757i)7-s + (−0.745 + 0.666i)8-s + (−0.300 − 0.144i)9-s + (0.599 + 0.995i)10-s + (−0.129 − 0.268i)11-s + (−0.557 − 0.149i)12-s + (0.193 + 0.401i)13-s + (−0.998 + 0.0559i)14-s + (−0.291 + 0.604i)15-s + (−0.997 − 0.0741i)16-s + (0.956 − 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.810 - 0.585i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.810 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652227 + 2.01765i\)
\(L(\frac12)\) \(\approx\) \(0.652227 + 2.01765i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.981 - 1.01i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (1.72 - 2.00i)T \)
good5 \( 1 + (-2.53 - 0.578i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (0.428 + 0.889i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.697 - 1.44i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-3.94 + 3.14i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 2.99T + 19T^{2} \)
23 \( 1 + (-2.65 - 2.11i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-0.154 - 0.193i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 1.95T + 31T^{2} \)
37 \( 1 + (-2.49 - 3.12i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (2.68 + 0.612i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-12.5 + 2.85i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (5.73 - 2.76i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-2.44 + 3.06i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-1.47 - 6.46i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (5.41 - 4.31i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 14.9iT - 67T^{2} \)
71 \( 1 + (-3.75 - 2.99i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.48 - 7.23i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 - 3.43iT - 79T^{2} \)
83 \( 1 + (-1.36 - 0.658i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (1.91 - 3.97i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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

−11.11825331545628654853413160972, −9.955165242292157912007953882836, −9.306099791219587311165659699269, −8.501690816331455740045426518982, −7.20145082620507538832046121309, −6.15746815556669457596939179576, −5.73905593772606729927713499672, −4.76299658835264010410474357785, −3.39638333140135333955500156911, −2.47350705064165521111194810016, 1.01458467790501064893362094506, 2.23566930246544991380044006561, 3.45055463336072215511421557718, 4.70245053869318520493814390656, 5.86483331245320726504458486432, 6.30866868506688503390686741601, 7.47779532275377671140967361254, 8.830543095417417776933435684317, 9.845683045729062812939713352585, 10.34476547410017865562692126391

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