↑

Properties

Label 2-3204-1068.863-c0-0-0
Degree $2$
Conductor $3204$
Sign $0.251 - 0.967i$
Analytic cond. $1.59900$
Root an. cond. $1.26451$
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  + (−0.989 − 0.142i)2-s + (0.959 + 0.281i)4-s + (−1.01 + 0.377i)5-s + (−0.909 − 0.415i)8-s + (1.05 − 0.229i)10-s + (−0.222 − 0.276i)13-s + (0.841 + 0.540i)16-s + (−0.0855 + 0.114i)17-s + (−1.07 + 0.0771i)20-s + (0.127 − 0.110i)25-s + (0.181 + 0.305i)26-s + (1.89 + 0.0677i)29-s + (−0.755 − 0.654i)32-s + (0.100 − 0.100i)34-s + (−0.942 + 0.390i)37-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (0.959 + 0.281i)4-s + (−1.01 + 0.377i)5-s + (−0.909 − 0.415i)8-s + (1.05 − 0.229i)10-s + (−0.222 − 0.276i)13-s + (0.841 + 0.540i)16-s + (−0.0855 + 0.114i)17-s + (−1.07 + 0.0771i)20-s + (0.127 − 0.110i)25-s + (0.181 + 0.305i)26-s + (1.89 + 0.0677i)29-s + (−0.755 − 0.654i)32-s + (0.100 − 0.100i)34-s + (−0.942 + 0.390i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.251 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5007757371\)
\(L(\frac12)\) \(\approx\) \(0.5007757371\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 \)
89 \( 1 + (0.349 - 0.936i)T \)
good5 \( 1 + (1.01 - 0.377i)T + (0.755 - 0.654i)T^{2} \)
7 \( 1 + (0.0713 - 0.997i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.222 + 0.276i)T + (-0.212 + 0.977i)T^{2} \)
17 \( 1 + (0.0855 - 0.114i)T + (-0.281 - 0.959i)T^{2} \)
19 \( 1 + (0.349 + 0.936i)T^{2} \)
23 \( 1 + (-0.936 + 0.349i)T^{2} \)
29 \( 1 + (-1.89 - 0.0677i)T + (0.997 + 0.0713i)T^{2} \)
31 \( 1 + (0.936 + 0.349i)T^{2} \)
37 \( 1 + (0.942 - 0.390i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.787 - 0.977i)T + (-0.212 - 0.977i)T^{2} \)
43 \( 1 + (0.997 - 0.0713i)T^{2} \)
47 \( 1 + (0.540 - 0.841i)T^{2} \)
53 \( 1 + (-1.05 - 0.574i)T + (0.540 + 0.841i)T^{2} \)
59 \( 1 + (0.977 - 0.212i)T^{2} \)
61 \( 1 + (-1.21 + 0.610i)T + (0.599 - 0.800i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.755 + 0.654i)T^{2} \)
73 \( 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.909 - 0.415i)T^{2} \)
83 \( 1 + (-0.877 - 0.479i)T^{2} \)
97 \( 1 + (0.398 - 0.871i)T + (-0.654 - 0.755i)T^{2} \)
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

−8.811384051910776152676734776697, −8.278411271164452545971046464172, −7.67035751195770484719526685699, −6.93317878816133872980539408429, −6.36456352534400381581431291594, −5.22798094000695159269247222529, −4.13938958375018297366391976553, −3.26281091451744074460541336042, −2.51397908864782321340244527841, −1.12960598976923335849497856481, 0.48183874894186913989365443863, 1.84152247081335107840735668282, 2.95079645231560273556145398706, 3.93817664257677161002679417597, 4.88434261240657152271601540320, 5.77364320687451359255742614411, 6.85927461966472648188527875912, 7.17770661876938779591502857720, 8.257296395573057903005368490361, 8.454121040557250902671355357846

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