↑

Properties

Label 2-396-132.131-c0-0-2
Degree $2$
Conductor $396$
Sign $0.816 - 0.577i$
Analytic cond. $0.197629$
Root an. cond. $0.444555$
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  + i·2-s − 4-s − 1.41i·5-s + 1.41·7-s − i·8-s + 1.41·10-s + i·11-s + 1.41i·14-s + 16-s − 1.41·19-s + 1.41i·20-s − 22-s − 1.00·25-s − 1.41·28-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 1.41i·5-s + 1.41·7-s − i·8-s + 1.41·10-s + i·11-s + 1.41i·14-s + 16-s − 1.41·19-s + 1.41i·20-s − 22-s − 1.00·25-s − 1.41·28-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(0.197629\)
Root analytic conductor: \(0.444555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :0),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8333333196\)
\(L(\frac12)\) \(\approx\) \(0.8333333196\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
11 \( 1 - iT \)
good5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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

−11.84811778468000534321762618270, −10.53546698480487644901203954389, −9.433538405310197988787395072462, −8.552842018811519360573732951411, −8.087823647879042209377343137560, −7.03174875722134841349653682731, −5.69116736876525591781948201456, −4.72417463360781465426598235763, −4.34634277242828612828933908640, −1.61885775541892759896728718420, 1.93204183660956945955701069077, 3.09401800829321625355083712985, 4.23878676859564224235093214418, 5.46440498461878671844368034694, 6.69381307210043456248616850124, 8.035827601218306045933111181729, 8.607991993189091257825554573210, 9.978029709582538840840246299421, 10.87648308423117055710174489632, 11.12723351507651080270080549041

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