↑

Properties

Label 2-1338-223.39-c1-0-1
Degree $2$
Conductor $1338$
Sign $-0.733 - 0.679i$
Analytic cond. $10.6839$
Root an. cond. $3.26863$
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  − 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.545 − 0.944i)5-s + (0.5 − 0.866i)6-s − 2.51·7-s − 8-s + (−0.499 − 0.866i)9-s + (0.545 + 0.944i)10-s + (−3.12 − 5.40i)11-s + (−0.5 + 0.866i)12-s + 5.01·13-s + 2.51·14-s + 1.09·15-s + 16-s + 1.49·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.243 − 0.422i)5-s + (0.204 − 0.353i)6-s − 0.951·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.172 + 0.298i)10-s + (−0.941 − 1.62i)11-s + (−0.144 + 0.249i)12-s + 1.39·13-s + 0.672·14-s + 0.281·15-s + 0.250·16-s + 0.361·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $-0.733 - 0.679i$
Analytic conductor: \(10.6839\)
Root analytic conductor: \(3.26863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1338} (931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1338,\ (\ :1/2),\ -0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2427561852\)
\(L(\frac12)\) \(\approx\) \(0.2427561852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.5 - 0.866i)T \)
223 \( 1 + (-13.6 + 6.06i)T \)
good5 \( 1 + (0.545 + 0.944i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.51T + 7T^{2} \)
11 \( 1 + (3.12 + 5.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 - 1.49T + 17T^{2} \)
19 \( 1 + (1.02 - 1.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.45 - 2.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 3.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.44 - 7.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.789 - 1.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.94T + 41T^{2} \)
43 \( 1 + (3.22 - 5.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.25 - 2.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.250 + 0.433i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.06T + 59T^{2} \)
61 \( 1 + (4.32 - 7.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.36 - 2.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.39 + 5.87i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.73 + 9.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.88 - 4.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.38 + 7.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.564 - 0.977i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.43 - 4.21i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.05863730636763209586308052097, −8.850384459078712239043845682983, −8.635144877101330008430784497615, −7.74368591044761135859427946575, −6.44859969980705833689609821214, −5.97318780824644603753073267000, −5.02782943918466067870759677442, −3.53756004352776756611567609501, −3.12941995274331602103616720881, −1.16866249369258726268556141755, 0.14520343448594345389414148950, 1.81814801232844255978167367225, 2.85093322708708023288032001219, 3.98077601963331026352303549045, 5.32236448801674666590854188656, 6.31575979245924040392342424062, 6.92624854345695781290071869777, 7.62554677554168060628985668886, 8.406414366122975733285189803128, 9.417714344935745335445468582922

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