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Properties

Label 2-396-132.131-c0-0-0
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

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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

\(L(\frac{1}{2})\) \(\approx\) \(0.6391055318\)
\(L(\frac12)\) \(\approx\) \(0.6391055318\)
\(L(1)\) 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 \( 1 + 2T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03336699095368667055852277436, −10.68134558946704579250995513263, −9.798766338136627439153410419324, −9.371498974573445726741266058374, −7.80556641984197528991770798090, −6.99592048379240584354034303385, −6.51822592798111987279895152199, −5.42277557530162217274634805283, −3.88636283930840579618451744476, −2.89721157293716921583638491800, 0.958307253214870246224113901070, 2.92329749882228784989150636523, 3.92914017525167207824548707956, 5.17653252046240449549977894351, 6.03456863959881782757908095565, 7.71703948220158806025498503962, 8.867223816082533822704324779708, 9.293812940388566929042688247174, 10.13959068817305069443380025314, 11.25658255722732749399499505687

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