↑

Properties

Label 2-1323-63.5-c1-0-4
Degree $2$
Conductor $1323$
Sign $-0.861 - 0.508i$
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  + 1.48i·2-s − 0.203·4-s + (0.154 + 0.267i)5-s + 2.66i·8-s + (−0.396 + 0.228i)10-s + (−2.73 − 1.58i)11-s + (3.00 + 1.73i)13-s − 4.36·16-s + (2.44 + 4.22i)17-s + (4.62 + 2.67i)19-s + (−0.0314 − 0.0544i)20-s + (2.34 − 4.06i)22-s + (−5.17 + 2.98i)23-s + (2.45 − 4.24i)25-s + (−2.57 + 4.45i)26-s + ⋯
L(s)  = 1  + 1.04i·2-s − 0.101·4-s + (0.0689 + 0.119i)5-s + 0.942i·8-s + (−0.125 + 0.0723i)10-s + (−0.825 − 0.476i)11-s + (0.833 + 0.481i)13-s − 1.09·16-s + (0.592 + 1.02i)17-s + (1.06 + 0.612i)19-s + (−0.00702 − 0.0121i)20-s + (0.500 − 0.866i)22-s + (−1.07 + 0.622i)23-s + (0.490 − 0.849i)25-s + (−0.504 + 0.874i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.861 - 0.508i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.861 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.668713523\)
\(L(\frac12)\) \(\approx\) \(1.668713523\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.48iT - 2T^{2} \)
5 \( 1 + (-0.154 - 0.267i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.73 + 1.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.00 - 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.44 - 4.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.62 - 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.17 - 2.98i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.70 - 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.52iT - 31T^{2} \)
37 \( 1 + (5.92 - 10.2i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.58 - 4.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 + (0.0740 - 0.0427i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 - 5.42iT - 61T^{2} \)
67 \( 1 + 0.110T + 67T^{2} \)
71 \( 1 + 7.78iT - 71T^{2} \)
73 \( 1 + (8.32 - 4.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + (-4.42 - 7.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.936 + 1.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 - 6.34i)T + (48.5 - 84.0i)T^{2} \)
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

−9.962727968024243317628258888544, −8.863178490049323522057260616144, −8.008766285597220250472965862680, −7.72805568679030685620365293203, −6.53861560779605379186533006696, −5.93866113104438018353266943603, −5.32425027786411831218736875671, −4.05534597780292782644335419850, −2.95328930075473802393408859207, −1.60975329536825823153588055147, 0.69003189293825921586900213711, 1.98306966036975292655755886006, 2.98579496209273933063182368763, 3.75131158937659299403034444985, 5.01129604381553607011959700675, 5.75607888353584585665595921300, 7.07919049509013130071563140252, 7.51873702194332303026125329678, 8.780209425240530442330806894897, 9.451880764457602445764912741810

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