↑

Properties

Label 2-644-644.643-c0-0-2
Degree $2$
Conductor $644$
Sign $1$
Analytic cond. $0.321397$
Root an. cond. $0.566919$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·5-s − 1.41·6-s + 7-s − 8-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 14-s − 2.00·15-s + 16-s + 1.41·17-s − 1.00·18-s − 1.41·20-s + 1.41·21-s + 23-s − 1.41·24-s + 1.00·25-s + 28-s + 2.00·30-s − 1.41·31-s − 32-s − 1.41·34-s − 1.41·35-s + 1.00·36-s + ⋯
L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·5-s − 1.41·6-s + 7-s − 8-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 14-s − 2.00·15-s + 16-s + 1.41·17-s − 1.00·18-s − 1.41·20-s + 1.41·21-s + 23-s − 1.41·24-s + 1.00·25-s + 28-s + 2.00·30-s − 1.41·31-s − 32-s − 1.41·34-s − 1.41·35-s + 1.00·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (643, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8558405730\)
\(L(\frac12)\) \(\approx\) \(0.8558405730\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 + 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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

−10.71151265954247160824932584218, −9.659033938005076050156005878228, −8.786175260830488425875697771192, −8.121857943294006454824228319259, −7.73792359421508395715617726156, −6.98784542127546217791065901538, −5.22167820167357457854019983097, −3.76390930507839525578237207628, −3.03562512031864936252004995522, −1.58755983396228822870147661188, 1.58755983396228822870147661188, 3.03562512031864936252004995522, 3.76390930507839525578237207628, 5.22167820167357457854019983097, 6.98784542127546217791065901538, 7.73792359421508395715617726156, 8.121857943294006454824228319259, 8.786175260830488425875697771192, 9.659033938005076050156005878228, 10.71151265954247160824932584218

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