↑

Properties

Label 2-3564-297.43-c0-0-0
Degree $2$
Conductor $3564$
Sign $-0.396 + 0.918i$
Analytic cond. $1.77866$
Root an. cond. $1.33366$
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  + (−1.76 − 0.642i)5-s + (0.939 − 0.342i)11-s + (0.173 + 0.984i)23-s + (1.93 + 1.62i)25-s + (−0.326 − 1.85i)31-s + (−0.766 − 1.32i)37-s + (0.326 − 1.85i)47-s + (−0.939 − 0.342i)49-s − 1.53·53-s − 1.87·55-s + (−1.76 − 0.642i)59-s + (0.266 − 0.223i)67-s + (0.173 + 0.300i)71-s + (1 − 1.73i)89-s + (−0.326 + 0.118i)97-s + ⋯
L(s)  = 1  + (−1.76 − 0.642i)5-s + (0.939 − 0.342i)11-s + (0.173 + 0.984i)23-s + (1.93 + 1.62i)25-s + (−0.326 − 1.85i)31-s + (−0.766 − 1.32i)37-s + (0.326 − 1.85i)47-s + (−0.939 − 0.342i)49-s − 1.53·53-s − 1.87·55-s + (−1.76 − 0.642i)59-s + (0.266 − 0.223i)67-s + (0.173 + 0.300i)71-s + (1 − 1.73i)89-s + (−0.326 + 0.118i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.396 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6498880823\)
\(L(\frac12)\) \(\approx\) \(0.6498880823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-0.939 + 0.342i)T \)
good5 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + 1.53T + T^{2} \)
59 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
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

−8.489167004678351576601293347545, −7.72061730379279031101876817694, −7.29665813728468000545737675013, −6.33881541886106782467001572030, −5.37242707436182873537522605785, −4.53437947876426313602034173953, −3.78934501307546850711404528282, −3.34422999630498510382144098094, −1.73393448992796831440264039771, −0.41736918382148976944233146607, 1.34055337713864657822510541913, 2.92161035760175058588819210955, 3.44776296308264238043640387964, 4.39752000734238587595061798943, 4.84364365036944699546311090429, 6.36918974264924351677738362277, 6.75152793543266415421898724342, 7.51927015185025789072423862749, 8.156430150356492855248046032265, 8.807702925020173349908020643699

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