↑

Properties

Label 2-588-4.3-c2-0-68
Degree $2$
Conductor $588$
Sign $0.243 + 0.969i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 − 0.245i)2-s − 1.73i·3-s + (3.87 − 0.973i)4-s − 0.424·5-s + (−0.424 − 3.43i)6-s + (7.46 − 2.88i)8-s − 2.99·9-s + (−0.842 + 0.104i)10-s − 6.71i·11-s + (−1.68 − 6.71i)12-s + 9.57·13-s + 0.735i·15-s + (14.1 − 7.55i)16-s + 15.4·17-s + (−5.95 + 0.735i)18-s − 2.85i·19-s + ⋯
L(s)  = 1  + (0.992 − 0.122i)2-s − 0.577i·3-s + (0.969 − 0.243i)4-s − 0.0848·5-s + (−0.0707 − 0.572i)6-s + (0.932 − 0.360i)8-s − 0.333·9-s + (−0.0842 + 0.0104i)10-s − 0.610i·11-s + (−0.140 − 0.559i)12-s + 0.736·13-s + 0.0490i·15-s + (0.881 − 0.472i)16-s + 0.909·17-s + (−0.330 + 0.0408i)18-s − 0.150i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.243 + 0.969i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.243 + 0.969i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.476548570\)
\(L(\frac12)\) \(\approx\) \(3.476548570\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 + 0.245i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + 0.424T + 25T^{2} \)
11 \( 1 + 6.71iT - 121T^{2} \)
13 \( 1 - 9.57T + 169T^{2} \)
17 \( 1 - 15.4T + 289T^{2} \)
19 \( 1 + 2.85iT - 361T^{2} \)
23 \( 1 + 32.6iT - 529T^{2} \)
29 \( 1 - 2.05T + 841T^{2} \)
31 \( 1 + 48.0iT - 961T^{2} \)
37 \( 1 - 18.5T + 1.36e3T^{2} \)
41 \( 1 + 24.5T + 1.68e3T^{2} \)
43 \( 1 - 81.2iT - 1.84e3T^{2} \)
47 \( 1 - 76.0iT - 2.20e3T^{2} \)
53 \( 1 - 64.9T + 2.80e3T^{2} \)
59 \( 1 + 46.0iT - 3.48e3T^{2} \)
61 \( 1 + 98.9T + 3.72e3T^{2} \)
67 \( 1 + 45.8iT - 4.48e3T^{2} \)
71 \( 1 - 114. iT - 5.04e3T^{2} \)
73 \( 1 + 11.0T + 5.32e3T^{2} \)
79 \( 1 + 7.63iT - 6.24e3T^{2} \)
83 \( 1 - 4.16iT - 6.88e3T^{2} \)
89 \( 1 - 159.T + 7.92e3T^{2} \)
97 \( 1 - 147.T + 9.40e3T^{2} \)
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

−10.63734406122720702978468530137, −9.551287078751306824282895711633, −8.213111937171068697486971704299, −7.58724061316576002260918424409, −6.29487305787235009492330738311, −5.93868211410373378009281593088, −4.62517132826342780221277891470, −3.53668948476083123870580045205, −2.46305617674908703394283515618, −1.02444723841606256603694103881, 1.73196454908239068486825708421, 3.28663764313178965723679114123, 3.96620008883965879536023654125, 5.15596620726013577994198917667, 5.80839859913507326139764497918, 6.99160354023736451684459064989, 7.82517836467878720476629631780, 8.910080173272152350993531515444, 10.09186267022432619722130033495, 10.66120782051826721532258157611

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