Properties

Degree 2
Conductor $ 2^{2} \cdot 11 $
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 − 5-s + 11-s + 15-s − 23-s + 27-s − 31-s − 33-s − 37-s + 2·47-s + 49-s + 2·53-s − 55-s − 59-s − 67-s + 69-s − 71-s − 81-s − 89-s + 93-s − 97-s + 2·103-s + 111-s − 113-s + 115-s + ⋯
L(s)  = 1  − 3-s − 5-s + 11-s + 15-s − 23-s + 27-s − 31-s − 33-s − 37-s + 2·47-s + 49-s + 2·53-s − 55-s − 59-s − 67-s + 69-s − 71-s − 81-s − 89-s + 93-s − 97-s + 2·103-s + 111-s − 113-s + 115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(44\)    =    \(2^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{44} (21, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 44,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3157625309\)
\(L(\frac12)\)  \(\approx\)  \(0.3157625309\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( ( 1 - T )^{2} \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T + T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.35967410986435713447548409957, −15.27319062813857186773037865264, −13.99851593145856548211536643059, −12.22485563619225556720266613180, −11.70308803032614155922865956844, −10.55715681078177439271694820025, −8.811336960685553603966194710597, −7.24907980722071614777800764527, −5.79558660706818170317355578628, −4.04199480857272067821281564908, 4.04199480857272067821281564908, 5.79558660706818170317355578628, 7.24907980722071614777800764527, 8.811336960685553603966194710597, 10.55715681078177439271694820025, 11.70308803032614155922865956844, 12.22485563619225556720266613180, 13.99851593145856548211536643059, 15.27319062813857186773037865264, 16.35967410986435713447548409957

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