↑

Properties

Label 2-3963-3963.1685-c0-0-0
Degree $2$
Conductor $3963$
Sign $-0.996 + 0.0863i$
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  + (0.142 − 0.989i)3-s − 4-s + (−0.424 − 1.94i)7-s + (−0.959 − 0.281i)9-s + (−0.142 + 0.989i)12-s + (1.59 − 1.19i)13-s + 16-s + (0.254 − 1.17i)19-s + (−1.98 + 0.142i)21-s + (−0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.424 + 1.94i)28-s + (0.540 + 0.158i)31-s + (0.959 + 0.281i)36-s + (0.989 + 1.14i)37-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s − 4-s + (−0.424 − 1.94i)7-s + (−0.959 − 0.281i)9-s + (−0.142 + 0.989i)12-s + (1.59 − 1.19i)13-s + 16-s + (0.254 − 1.17i)19-s + (−1.98 + 0.142i)21-s + (−0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.424 + 1.94i)28-s + (0.540 + 0.158i)31-s + (0.959 + 0.281i)36-s + (0.989 + 1.14i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3963\)    =    \(3 \cdot 1321\)
Sign: $-0.996 + 0.0863i$
Analytic conductor: \(1.97779\)
Root analytic conductor: \(1.40634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3963} (1685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3963,\ (\ :0),\ -0.996 + 0.0863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9346497334\)
\(L(\frac12)\) \(\approx\) \(0.9346497334\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.142 + 0.989i)T \)
1321 \( 1 + (-0.654 + 0.755i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.424 + 1.94i)T + (-0.909 + 0.415i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (-1.59 + 1.19i)T + (0.281 - 0.959i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.254 + 1.17i)T + (-0.909 - 0.415i)T^{2} \)
23 \( 1 + (-0.989 + 0.142i)T^{2} \)
29 \( 1 + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.540 - 0.158i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.989 - 1.14i)T + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.153 + 1.07i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.540 - 0.841i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.755 - 0.654i)T^{2} \)
61 \( 1 + (-1.83 - 0.682i)T + (0.755 + 0.654i)T^{2} \)
67 \( 1 + (-1.75 + 0.125i)T + (0.989 - 0.142i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (1.54 - 0.841i)T + (0.540 - 0.841i)T^{2} \)
79 \( 1 + (1.86 + 0.697i)T + (0.755 + 0.654i)T^{2} \)
83 \( 1 + (0.909 - 0.415i)T^{2} \)
89 \( 1 + (-0.281 + 0.959i)T^{2} \)
97 \( 1 + (0.203 + 0.373i)T + (-0.540 + 0.841i)T^{2} \)
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

−8.246593565964173762446710038122, −7.61839021824679133947891595547, −6.88529435415487309051945539919, −6.19614992513378285095028446238, −5.36875230414588537134641774738, −4.29522511994976256971578928190, −3.65870493191489085330306290516, −2.91961052759008848612312495098, −1.17378577085512078214747068893, −0.63442282009521043948500375947, 1.77199356781637763170879242497, 2.92590846624487176154947023348, 3.76850032790786255650523119445, 4.30142094014743589530958840312, 5.44620600034448025923538395094, 5.74423061780805801661155056619, 6.42295711053266479442752513470, 8.042685556867150634239989512889, 8.465657748800392840499658606743, 9.046725075231146163273262281004

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