↑

Properties

Label 2-1323-63.58-c1-0-9
Degree $2$
Conductor $1323$
Sign $-0.590 - 0.807i$
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  − 1.69·2-s + 0.888·4-s + (1.79 + 3.10i)5-s + 1.88·8-s + (−3.04 − 5.28i)10-s + (−1.40 + 2.43i)11-s + (0.5 − 0.866i)13-s − 4.98·16-s + (2.05 + 3.56i)17-s + (−0.444 + 0.769i)19-s + (1.59 + 2.76i)20-s + (2.38 − 4.13i)22-s + (2.93 + 5.08i)23-s + (−3.93 + 6.82i)25-s + (−0.849 + 1.47i)26-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.444·4-s + (0.802 + 1.38i)5-s + 0.667·8-s + (−0.964 − 1.67i)10-s + (−0.423 + 0.733i)11-s + (0.138 − 0.240i)13-s − 1.24·16-s + (0.498 + 0.863i)17-s + (−0.101 + 0.176i)19-s + (0.356 + 0.617i)20-s + (0.509 − 0.882i)22-s + (0.612 + 1.06i)23-s + (−0.787 + 1.36i)25-s + (−0.166 + 0.288i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.590 - 0.807i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8211911215\)
\(L(\frac12)\) \(\approx\) \(0.8211911215\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69T + 2T^{2} \)
5 \( 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.05 - 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.444 - 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.93 - 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.849 + 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.98T + 31T^{2} \)
37 \( 1 + (2.38 - 4.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.70 + 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.66T + 47T^{2} \)
53 \( 1 + (0.0618 + 0.107i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.87T + 59T^{2} \)
61 \( 1 + 3.87T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 + (5.32 + 9.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + (-2.05 - 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.80 - 8.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.945604319821775548636555222289, −9.334115246666865953805535159009, −8.271786576130607640449707860534, −7.57238693415292751789874737288, −6.86677367236225550459336347235, −6.04859590140116247084695255465, −4.98877602546986830548731473105, −3.61063158211119605973804945418, −2.47579428703840871778787577310, −1.50135411808745831132655887918, 0.55770895285854709590045493652, 1.45180098131268483888199772694, 2.76513015153184071339578959636, 4.47438997810788147694569975501, 5.08144259704426812142944922539, 6.05957871843619356938303999539, 7.11133939099677762733120390355, 8.224099465232533369911917912350, 8.536550188975391711020078838653, 9.360484662635537100473894792788

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