↑

Properties

Label 2-1323-63.41-c1-0-4
Degree $2$
Conductor $1323$
Sign $-0.454 - 0.890i$
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  + (1.30 − 0.755i)2-s + (0.140 − 0.242i)4-s + (−0.387 + 0.671i)5-s + 2.59i·8-s + 1.17i·10-s + (−3.32 + 1.92i)11-s + (−2.54 − 1.46i)13-s + (2.24 + 3.88i)16-s − 5.39·17-s − 0.434i·19-s + (0.108 + 0.188i)20-s + (−2.90 + 5.02i)22-s + (−0.0482 − 0.0278i)23-s + (2.19 + 3.80i)25-s − 4.43·26-s + ⋯
L(s)  = 1  + (0.924 − 0.533i)2-s + (0.0700 − 0.121i)4-s + (−0.173 + 0.300i)5-s + 0.918i·8-s + 0.370i·10-s + (−1.00 + 0.579i)11-s + (−0.705 − 0.407i)13-s + (0.560 + 0.970i)16-s − 1.30·17-s − 0.0997i·19-s + (0.0243 + 0.0421i)20-s + (−0.618 + 1.07i)22-s + (−0.0100 − 0.00580i)23-s + (0.439 + 0.761i)25-s − 0.869·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.089118898\)
\(L(\frac12)\) \(\approx\) \(1.089118898\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 0.755i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.387 - 0.671i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.32 - 1.92i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.54 + 1.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.39T + 17T^{2} \)
19 \( 1 + 0.434iT - 19T^{2} \)
23 \( 1 + (0.0482 + 0.0278i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.187 + 0.108i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.67 + 3.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.29T + 37T^{2} \)
41 \( 1 + (3.78 - 6.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.42 - 11.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.482 + 0.836i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.46iT - 53T^{2} \)
59 \( 1 + (-1.56 + 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.01 + 1.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.10 + 3.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 - 8.15iT - 73T^{2} \)
79 \( 1 + (-2.48 - 4.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.31 - 7.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (-1.24 + 0.716i)T + (48.5 - 84.0i)T^{2} \)
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

−10.07196342491204693468236534151, −9.153333676246419041502567891407, −8.156747623713407975026244120878, −7.47974015819250172243284142767, −6.54146296767104631042105160715, −5.28637836515394085694035587089, −4.83779200343909020104099159175, −3.82297215017257889320591073771, −2.83439915738273281591983416215, −2.08765291885010739722953212274, 0.30301554613614453515666454133, 2.20417325550207367601577295533, 3.47471078395543502649846844077, 4.45176830421559332250353205873, 5.11932195099776566908515309844, 5.85542474449847291692321238516, 6.85148688451954684574208524147, 7.43154463313862531970121347790, 8.619895545004433629318285506429, 9.165487656283680168947254757142

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