↑

Properties

Label 2-1323-63.25-c1-0-15
Degree $2$
Conductor $1323$
Sign $0.755 - 0.655i$
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  − 2.46·2-s + 4.05·4-s + (−1.82 + 3.16i)5-s − 5.05·8-s + (4.50 − 7.79i)10-s + (0.203 + 0.351i)11-s + (0.243 + 0.421i)13-s + 4.32·16-s + (2.42 − 4.20i)17-s + (0.986 + 1.70i)19-s + (−7.41 + 12.8i)20-s + (−0.5 − 0.866i)22-s + (2.32 − 4.02i)23-s + (−4.19 − 7.25i)25-s + (−0.598 − 1.03i)26-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.02·4-s + (−0.817 + 1.41i)5-s − 1.78·8-s + (1.42 − 2.46i)10-s + (0.0612 + 0.106i)11-s + (0.0675 + 0.116i)13-s + 1.08·16-s + (0.588 − 1.01i)17-s + (0.226 + 0.392i)19-s + (−1.65 + 2.87i)20-s + (−0.106 − 0.184i)22-s + (0.484 − 0.839i)23-s + (−0.838 − 1.45i)25-s + (−0.117 − 0.203i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5940410367\)
\(L(\frac12)\) \(\approx\) \(0.5940410367\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 2T^{2} \)
5 \( 1 + (1.82 - 3.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.203 - 0.351i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.243 - 0.421i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.42 + 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.986 - 1.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.82 + 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 + (1.16 + 2.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.16 + 2.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.31T + 47T^{2} \)
53 \( 1 + (1.78 - 3.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.11T + 59T^{2} \)
61 \( 1 + 8.02T + 61T^{2} \)
67 \( 1 - 3.60T + 67T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + (0.986 - 1.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.16T + 79T^{2} \)
83 \( 1 + (-6.08 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.41 - 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.74 + 8.21i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.765799488373076647358086207542, −8.958739317588466764678731280828, −7.893039689443258085162152047261, −7.63621634385719331881944066000, −6.75660061059189716999409479639, −6.17193309644191933255635344889, −4.47181133495484690327980358920, −3.11494114235324957303412933576, −2.44165767051251130760514809354, −0.76341914505706449210158371551, 0.74346174352306033589645740506, 1.54789674998938398771354830822, 3.16376671796952848201906829704, 4.41207707529591522441895164539, 5.44604929935622401441513242510, 6.58089463178952441647479530431, 7.61601272953496087011284455057, 8.043236528300539195994432326499, 8.864115057472261874704374874123, 9.165860564865015270840239530975

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