↑

Properties

Label 2-3204-1068.515-c0-0-0
Degree $2$
Conductor $3204$
Sign $-0.00388 - 0.999i$
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  + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.948 − 1.73i)5-s + (−0.281 + 0.959i)8-s + (1.58 + 1.18i)10-s + (−0.769 + 1.53i)13-s + (−0.142 − 0.989i)16-s + (−0.398 + 0.148i)17-s + (−1.93 − 0.420i)20-s + (−1.57 + 2.45i)25-s + (0.0613 − 1.71i)26-s + (0.175 + 1.63i)29-s + (0.540 + 0.841i)32-s + (0.300 − 0.300i)34-s + (0.0659 − 0.0273i)37-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.948 − 1.73i)5-s + (−0.281 + 0.959i)8-s + (1.58 + 1.18i)10-s + (−0.769 + 1.53i)13-s + (−0.142 − 0.989i)16-s + (−0.398 + 0.148i)17-s + (−1.93 − 0.420i)20-s + (−1.57 + 2.45i)25-s + (0.0613 − 1.71i)26-s + (0.175 + 1.63i)29-s + (0.540 + 0.841i)32-s + (0.300 − 0.300i)34-s + (0.0659 − 0.0273i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $-0.00388 - 0.999i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ -0.00388 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3383378551\)
\(L(\frac12)\) \(\approx\) \(0.3383378551\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 0.415i)T \)
3 \( 1 \)
89 \( 1 + (0.877 + 0.479i)T \)
good5 \( 1 + (0.948 + 1.73i)T + (-0.540 + 0.841i)T^{2} \)
7 \( 1 + (0.212 + 0.977i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.769 - 1.53i)T + (-0.599 - 0.800i)T^{2} \)
17 \( 1 + (0.398 - 0.148i)T + (0.755 - 0.654i)T^{2} \)
19 \( 1 + (0.877 - 0.479i)T^{2} \)
23 \( 1 + (0.479 + 0.877i)T^{2} \)
29 \( 1 + (-0.175 - 1.63i)T + (-0.977 + 0.212i)T^{2} \)
31 \( 1 + (-0.479 + 0.877i)T^{2} \)
37 \( 1 + (-0.0659 + 0.0273i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.400 + 0.800i)T + (-0.599 + 0.800i)T^{2} \)
43 \( 1 + (-0.977 - 0.212i)T^{2} \)
47 \( 1 + (-0.989 + 0.142i)T^{2} \)
53 \( 1 + (0.133 - 1.86i)T + (-0.989 - 0.142i)T^{2} \)
59 \( 1 + (-0.800 - 0.599i)T^{2} \)
61 \( 1 + (1.23 - 0.222i)T + (0.936 - 0.349i)T^{2} \)
67 \( 1 + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.540 - 0.841i)T^{2} \)
73 \( 1 + (-1.29 + 0.186i)T + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.281 + 0.959i)T^{2} \)
83 \( 1 + (0.0713 - 0.997i)T^{2} \)
97 \( 1 + (-1.91 - 0.562i)T + (0.841 + 0.540i)T^{2} \)
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

−9.079935103923378440947013522153, −8.457317715751656422781985686694, −7.58881361916844093922047655242, −7.11396037247816831733227864674, −6.14699679410003416361701756785, −5.02728897385685894250991740882, −4.72072617613003724541588279715, −3.67131824582080156285785076252, −2.08233081754813345799929464247, −1.17716716340565036932901482512, 0.29879257259264496760134103691, 2.25227152926214226944138729871, 2.93903444322738039012534116400, 3.52163764137533449165340328995, 4.54455454291613152538618342830, 5.99787717178848142366911660071, 6.66150651924731933831641733598, 7.42953307680060592131877033899, 7.87107672946513347769211140149, 8.411085775169118753856341082316

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