Properties

Degree $2$
Conductor $384$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 2·11-s − 25-s − 27-s − 2·33-s − 49-s − 2·59-s − 2·73-s + 75-s + 81-s + 2·83-s + 2·97-s + 2·99-s − 2·107-s + ⋯
L(s)  = 1  − 3-s + 9-s + 2·11-s − 25-s − 27-s − 2·33-s − 49-s − 2·59-s − 2·73-s + 75-s + 81-s + 2·83-s + 2·97-s + 2·99-s − 2·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6758275560\)
\(L(\frac12)\) \(\approx\) \(0.6758275560\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 + T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−11.70957848674615572709908029269, −10.80323671458423038130760524037, −9.755549068104677975073807155194, −9.033119266802489769096367800822, −7.64700335958132785483447087494, −6.60071251389104815841797533111, −5.97981258635214669939844358455, −4.66487465990651944039652242126, −3.71113239212401521902242330066, −1.53070354608733596024417520270, 1.53070354608733596024417520270, 3.71113239212401521902242330066, 4.66487465990651944039652242126, 5.97981258635214669939844358455, 6.60071251389104815841797533111, 7.64700335958132785483447087494, 9.033119266802489769096367800822, 9.755549068104677975073807155194, 10.80323671458423038130760524037, 11.70957848674615572709908029269

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