↑

Properties

Label 2-333-37.31-c2-0-13
Degree $2$
Conductor $333$
Sign $-0.997 - 0.0655i$
Analytic cond. $9.07359$
Root an. cond. $3.01224$
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  + (2.74 + 2.74i)2-s + 11.0i·4-s + (−0.592 + 0.592i)5-s − 2.28·7-s + (−19.4 + 19.4i)8-s − 3.25·10-s + 8.29i·11-s + (3.30 − 3.30i)13-s + (−6.28 − 6.28i)14-s − 62.5·16-s + (5.43 − 5.43i)17-s + (15.5 − 15.5i)19-s + (−6.57 − 6.57i)20-s + (−22.7 + 22.7i)22-s + (−15.0 + 15.0i)23-s + ⋯
L(s)  = 1  + (1.37 + 1.37i)2-s + 2.77i·4-s + (−0.118 + 0.118i)5-s − 0.326·7-s + (−2.43 + 2.43i)8-s − 0.325·10-s + 0.754i·11-s + (0.253 − 0.253i)13-s + (−0.448 − 0.448i)14-s − 3.91·16-s + (0.319 − 0.319i)17-s + (0.820 − 0.820i)19-s + (−0.328 − 0.328i)20-s + (−1.03 + 1.03i)22-s + (−0.652 + 0.652i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $-0.997 - 0.0655i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ -0.997 - 0.0655i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0967841 + 2.94930i\)
\(L(\frac12)\) \(\approx\) \(0.0967841 + 2.94930i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 + (-29.7 + 22.0i)T \)
good2 \( 1 + (-2.74 - 2.74i)T + 4iT^{2} \)
5 \( 1 + (0.592 - 0.592i)T - 25iT^{2} \)
7 \( 1 + 2.28T + 49T^{2} \)
11 \( 1 - 8.29iT - 121T^{2} \)
13 \( 1 + (-3.30 + 3.30i)T - 169iT^{2} \)
17 \( 1 + (-5.43 + 5.43i)T - 289iT^{2} \)
19 \( 1 + (-15.5 + 15.5i)T - 361iT^{2} \)
23 \( 1 + (15.0 - 15.0i)T - 529iT^{2} \)
29 \( 1 + (4.60 + 4.60i)T + 841iT^{2} \)
31 \( 1 + (-36.3 - 36.3i)T + 961iT^{2} \)
41 \( 1 + 4.50iT - 1.68e3T^{2} \)
43 \( 1 + (-24.2 + 24.2i)T - 1.84e3iT^{2} \)
47 \( 1 - 29.6T + 2.20e3T^{2} \)
53 \( 1 - 23.5T + 2.80e3T^{2} \)
59 \( 1 + (-56.8 + 56.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-43.1 - 43.1i)T + 3.72e3iT^{2} \)
67 \( 1 - 109. iT - 4.48e3T^{2} \)
71 \( 1 + 46.2T + 5.04e3T^{2} \)
73 \( 1 + 19.8iT - 5.32e3T^{2} \)
79 \( 1 + (-66.5 + 66.5i)T - 6.24e3iT^{2} \)
83 \( 1 + 46.6T + 6.88e3T^{2} \)
89 \( 1 + (42.1 + 42.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (86.1 - 86.1i)T - 9.40e3iT^{2} \)
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

−12.12442283214283446347939963978, −11.36437683766800944958656980495, −9.742478950311596497190839854436, −8.613609754590708583583367678502, −7.49335305514293139243303066834, −6.99804735179174592957974260481, −5.84467368365651611501070209691, −5.02040363527066936585769081500, −3.90992392186341098813243938827, −2.85245078022598534586621163376, 0.929530584339554853115567489368, 2.49333288389959006305577523856, 3.60255244490501236607896458375, 4.47617142550722284366465890794, 5.75616118010211440815334436661, 6.37317668584526486742707472812, 8.202047916678402721276033282735, 9.585722646156990176461919217280, 10.22196589431297496096458410296, 11.19356768931984627242705954633

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