↑

Properties

Label 2-3963-3963.1241-c0-0-0
Degree $2$
Conductor $3963$
Sign $-0.881 + 0.471i$
Analytic cond. $1.97779$
Root an. cond. $1.40634$
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  + (−0.841 + 0.540i)3-s − 4-s + (−0.0683 + 0.956i)7-s + (0.415 − 0.909i)9-s + (0.841 − 0.540i)12-s + (−0.936 − 0.203i)13-s + 16-s + (0.0303 + 0.424i)19-s + (−0.459 − 0.841i)21-s + (0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (0.0683 − 0.956i)28-s + (−0.755 + 1.65i)31-s + (−0.415 + 0.909i)36-s + (−0.540 + 0.158i)37-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s − 4-s + (−0.0683 + 0.956i)7-s + (0.415 − 0.909i)9-s + (0.841 − 0.540i)12-s + (−0.936 − 0.203i)13-s + 16-s + (0.0303 + 0.424i)19-s + (−0.459 − 0.841i)21-s + (0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (0.0683 − 0.956i)28-s + (−0.755 + 1.65i)31-s + (−0.415 + 0.909i)36-s + (−0.540 + 0.158i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3963\)    =    \(3 \cdot 1321\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(1.97779\)
Root analytic conductor: \(1.40634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3963} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3963,\ (\ :0),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1478744982\)
\(L(\frac12)\) \(\approx\) \(0.1478744982\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.841 - 0.540i)T \)
1321 \( 1 + (-0.959 - 0.281i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.0683 - 0.956i)T + (-0.989 - 0.142i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.936 + 0.203i)T + (0.909 + 0.415i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.0303 - 0.424i)T + (-0.989 + 0.142i)T^{2} \)
23 \( 1 + (0.540 - 0.841i)T^{2} \)
29 \( 1 + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.755 - 1.65i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (1.27 + 0.817i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.755 + 0.654i)T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.281 - 0.959i)T^{2} \)
61 \( 1 + (1.19 + 1.59i)T + (-0.281 + 0.959i)T^{2} \)
67 \( 1 + (-0.334 - 0.613i)T + (-0.540 + 0.841i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (0.244 + 0.654i)T + (-0.755 + 0.654i)T^{2} \)
79 \( 1 + (0.574 + 0.767i)T + (-0.281 + 0.959i)T^{2} \)
83 \( 1 + (0.989 + 0.142i)T^{2} \)
89 \( 1 + (-0.909 - 0.415i)T^{2} \)
97 \( 1 + (-0.133 + 0.0498i)T + (0.755 - 0.654i)T^{2} \)
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

−9.004773643789647104268733986736, −8.672284086217218497857144413006, −7.58695394695747111505195829042, −6.75551154999756616432673704145, −5.85236654869555390861625834151, −5.06267237368561791084378576634, −4.97359711770150114333439925760, −3.75816257638791888602756327612, −3.05960341661638957760040504677, −1.54875134103665473940541502029, 0.10771235320910474944247970694, 1.22820177935477053944825809094, 2.53872177830580782754273048287, 3.85245419589646270690436737217, 4.54394036267983996247339764333, 5.11171526483762805936434494480, 5.96561302600173234196885229627, 6.81677310563852292810370550387, 7.45445355052169312394043431006, 8.033107175837647397103588456144

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