↑

Properties

Label 2-592-37.33-c1-0-9
Degree $2$
Conductor $592$
Sign $0.886 - 0.462i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
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  + (0.260 + 1.47i)3-s + (−1.75 + 1.47i)5-s + (3.54 − 2.97i)7-s + (0.709 − 0.258i)9-s + (2.89 − 5.02i)11-s + (1.41 + 0.513i)13-s + (−2.62 − 2.20i)15-s + (−0.343 + 0.125i)17-s + (0.769 + 4.36i)19-s + (5.30 + 4.45i)21-s + (−1.15 − 1.99i)23-s + (0.0400 − 0.227i)25-s + (2.81 + 4.87i)27-s + (−1.46 + 2.53i)29-s − 0.428·31-s + ⋯
L(s)  = 1  + (0.150 + 0.851i)3-s + (−0.783 + 0.657i)5-s + (1.33 − 1.12i)7-s + (0.236 − 0.0860i)9-s + (0.874 − 1.51i)11-s + (0.391 + 0.142i)13-s + (−0.677 − 0.568i)15-s + (−0.0833 + 0.0303i)17-s + (0.176 + 1.00i)19-s + (1.15 + 0.971i)21-s + (−0.240 − 0.416i)23-s + (0.00800 − 0.0454i)25-s + (0.541 + 0.937i)27-s + (−0.271 + 0.470i)29-s − 0.0769·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65033 + 0.404710i\)
\(L(\frac12)\) \(\approx\) \(1.65033 + 0.404710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (3.07 - 5.24i)T \)
good3 \( 1 + (-0.260 - 1.47i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (1.75 - 1.47i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-3.54 + 2.97i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-2.89 + 5.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.41 - 0.513i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.343 - 0.125i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.769 - 4.36i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (1.15 + 1.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.46 - 2.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.428T + 31T^{2} \)
41 \( 1 + (-0.112 - 0.0408i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 + (3.30 + 5.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.42 + 6.22i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (6.22 + 5.22i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.40 - 0.874i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (10.8 - 9.07i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.40 + 7.98i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + (8.48 - 7.12i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.92 + 2.52i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-10.7 - 9.02i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.83 + 4.90i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.96528262040743638331547300273, −10.10526800520036899401646909114, −8.918650728172382945064342775346, −8.123743462507701720131701681577, −7.33263322375550131581649491649, −6.28734287629448944066204213068, −4.90716536626528527290024752003, −3.83909386075898937713263385289, −3.56163704490602141431086063574, −1.27632098674875133389341141089, 1.37575179012117683143608810099, 2.29470563041123978010234298254, 4.26970399755972335178668226877, 4.84189077166854722393917075636, 6.13142598495861545794251354932, 7.41465163364718690344700095397, 7.77624822038151839828244989448, 8.799809107682927273142670497576, 9.401218492750010302869322535414, 10.89461186624953965378094864298

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