↑

Properties

Label 2-1323-9.4-c1-0-35
Degree $2$
Conductor $1323$
Sign $-0.512 - 0.858i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  + (−0.863 − 1.49i)2-s + (−0.490 + 0.849i)4-s + (1.75 − 3.04i)5-s − 1.75·8-s − 6.06·10-s + (−3.04 − 5.27i)11-s + (0.560 − 0.970i)13-s + (2.49 + 4.32i)16-s − 1.20·17-s − 2.20·19-s + (1.72 + 2.98i)20-s + (−5.25 + 9.10i)22-s + (−0.636 + 1.10i)23-s + (−3.66 − 6.35i)25-s − 1.93·26-s + ⋯
L(s)  = 1  + (−0.610 − 1.05i)2-s + (−0.245 + 0.424i)4-s + (0.785 − 1.36i)5-s − 0.621·8-s − 1.91·10-s + (−0.918 − 1.59i)11-s + (0.155 − 0.269i)13-s + (0.624 + 1.08i)16-s − 0.292·17-s − 0.505·19-s + (0.385 + 0.667i)20-s + (−1.12 + 1.94i)22-s + (−0.132 + 0.229i)23-s + (−0.733 − 1.27i)25-s − 0.379·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.512 - 0.858i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8536908494\)
\(L(\frac12)\) \(\approx\) \(0.8536908494\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.863 + 1.49i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.75 + 3.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.04 + 5.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.560 + 0.970i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
19 \( 1 + 2.20T + 19T^{2} \)
23 \( 1 + (0.636 - 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0942 + 0.163i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.57T + 37T^{2} \)
41 \( 1 + (1.68 - 2.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.86 + 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.33T + 53T^{2} \)
59 \( 1 + (5.63 - 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.95 + 6.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 5.31T + 73T^{2} \)
79 \( 1 + (4.60 + 7.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.624 + 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.54T + 89T^{2} \)
97 \( 1 + (-8.24 - 14.2i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.036957346711273692915993747730, −8.661619980951899269271151278845, −7.998350730808178977402180737845, −6.36164890117946918053578964423, −5.65815923092300914254256382873, −4.92391387920954942792791872194, −3.52410875043053590586685666227, −2.52468938598432816941492101816, −1.41936647326665627097840450284, −0.42310189303683707110642045448, 2.15213351011768435848861527696, 2.85455581404530794243877140795, 4.36590205727251952022405633585, 5.53791065872066938084001390390, 6.39612209833652518228462726717, 6.88914504487346815363812140602, 7.57663469404567269731247997322, 8.335640226036460348798951888846, 9.449668306282518543860925011490, 9.965617365238899925693095914131

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