Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 4·7-s + 9-s − 2·13-s − 2·15-s + 2·17-s − 4·21-s − 8·23-s − 25-s − 27-s − 6·29-s + 8·31-s + 8·35-s − 6·37-s + 2·39-s + 2·41-s + 2·45-s − 8·47-s + 9·49-s − 2·51-s − 6·53-s − 4·59-s + 6·61-s + 4·63-s − 4·65-s − 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 0.485·17-s − 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 1.35·35-s − 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.496·65-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23232\)    =    \(2^{6} \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{23232} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 23232,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.71060189486686, −15.19317952419076, −14.48083874421968, −14.16123551433239, −13.72725420203288, −13.05682403020026, −12.31236507607775, −11.94708425263658, −11.36587717357382, −10.92134134226120, −10.07618836103668, −9.940357485826527, −9.234182412938210, −8.308276373897102, −8.004676901368224, −7.372679105434285, −6.609857978524545, −5.916201310220661, −5.508081728361655, −4.877894537751575, −4.370896262537021, −3.539043217157536, −2.411679509353098, −1.839477930240790, −1.277437264372568, 0, 1.277437264372568, 1.839477930240790, 2.411679509353098, 3.539043217157536, 4.370896262537021, 4.877894537751575, 5.508081728361655, 5.916201310220661, 6.609857978524545, 7.372679105434285, 8.004676901368224, 8.308276373897102, 9.234182412938210, 9.940357485826527, 10.07618836103668, 10.92134134226120, 11.36587717357382, 11.94708425263658, 12.31236507607775, 13.05682403020026, 13.72725420203288, 14.16123551433239, 14.48083874421968, 15.19317952419076, 15.71060189486686

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