↑

Properties

Label 2-63e2-9.5-c0-0-6
Degree $2$
Conductor $3969$
Sign $0.819 + 0.573i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.448 + 0.258i)2-s + (−0.366 + 0.633i)4-s − 0.896i·8-s + (1.67 − 0.965i)11-s + (−0.133 − 0.232i)16-s + (−0.500 + 0.866i)22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + (1.22 − 0.707i)29-s + (0.896 + 0.517i)32-s − 1.73·37-s + (−0.866 − 1.5i)43-s + 1.41i·44-s + 0.732·46-s + (0.448 + 0.258i)50-s + ⋯
L(s)  = 1  + (−0.448 + 0.258i)2-s + (−0.366 + 0.633i)4-s − 0.896i·8-s + (1.67 − 0.965i)11-s + (−0.133 − 0.232i)16-s + (−0.500 + 0.866i)22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + (1.22 − 0.707i)29-s + (0.896 + 0.517i)32-s − 1.73·37-s + (−0.866 − 1.5i)43-s + 1.41i·44-s + 0.732·46-s + (0.448 + 0.258i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.819 + 0.573i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2402, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.819 + 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8477918462\)
\(L(\frac12)\) \(\approx\) \(0.8477918462\)
\(L(1)\) 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 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.93iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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

−8.385470264595209671142209801942, −8.237645013807945543925275271263, −6.98246873226578366144171423224, −6.57916332452442307471428797826, −5.80864556962423157329483888029, −4.61353917208275076085172978257, −3.88197138031638689728588961451, −3.34083136747838564926186375780, −2.01718718337560051501795579784, −0.60559291233683419718133442488, 1.36503174113367518750947159418, 1.85334597071646929283288854173, 3.28093912669878863784203887205, 4.23201243900622063641648084099, 4.86257874089898903256049191863, 5.80958416213657755846038241840, 6.48754979548059073902629156622, 7.24852598956582805701942972711, 8.129361617930492903575990050059, 8.946234966296008176312904784662

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