↑

Properties

Label 2-1323-63.20-c1-0-2
Degree $2$
Conductor $1323$
Sign $0.141 - 0.989i$
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  + (−0.621 − 0.359i)2-s + (−0.742 − 1.28i)4-s + (−0.723 − 1.25i)5-s + 2.50i·8-s + 1.03i·10-s + (−1.55 − 0.900i)11-s + (1.88 − 1.09i)13-s + (−0.585 + 1.01i)16-s − 3.90·17-s + 4.01i·19-s + (−1.07 + 1.86i)20-s + (0.646 + 1.11i)22-s + (−4.91 + 2.83i)23-s + (1.45 − 2.51i)25-s − 1.56·26-s + ⋯
L(s)  = 1  + (−0.439 − 0.253i)2-s + (−0.371 − 0.642i)4-s + (−0.323 − 0.560i)5-s + 0.884i·8-s + 0.328i·10-s + (−0.470 − 0.271i)11-s + (0.523 − 0.302i)13-s + (−0.146 + 0.253i)16-s − 0.947·17-s + 0.920i·19-s + (−0.240 + 0.416i)20-s + (0.137 + 0.238i)22-s + (−1.02 + 0.591i)23-s + (0.290 − 0.503i)25-s − 0.307·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.141 - 0.989i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2707968797\)
\(L(\frac12)\) \(\approx\) \(0.2707968797\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.621 + 0.359i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.723 + 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.55 + 0.900i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.88 + 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.90T + 17T^{2} \)
19 \( 1 - 4.01iT - 19T^{2} \)
23 \( 1 + (4.91 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.49 + 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.45 + 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.823T + 37T^{2} \)
41 \( 1 + (-5.90 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.16 - 2.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.15iT - 53T^{2} \)
59 \( 1 + (-4.89 - 8.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.03 + 1.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.156 - 0.270i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 - 2.80iT - 73T^{2} \)
79 \( 1 + (6.21 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.60 + 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-13.4 - 7.75i)T + (48.5 + 84.0i)T^{2} \)
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

−9.765706351590656627370907648386, −9.098061776583282154310695362732, −8.173358818825569390048895848299, −7.84906030069760428417684979902, −6.26603560308179519965404469542, −5.72487027931302269179386209193, −4.68965050355222038548497808795, −3.89835657929048395383971872066, −2.40265933344758858120341975042, −1.20262340592666635228558831520, 0.14282531916029888967308713593, 2.18128342847929387094672845631, 3.38762392313402495790614579589, 4.14847942148638283259909536995, 5.17209759064895115591665831379, 6.47766203719820160122249787707, 7.13112417464123038145068926144, 7.75867404020333355693172000430, 8.774928632255511725499358424807, 9.121717892016648418348484935123

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