Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 7-s + 9-s − 2·10-s + 11-s + 2·12-s − 6·13-s − 2·14-s + 15-s − 4·16-s − 7·17-s − 2·18-s − 5·19-s + 2·20-s + 21-s − 2·22-s − 23-s + 25-s + 12·26-s + 27-s + 2·28-s − 5·29-s − 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s − 16-s − 1.69·17-s − 0.471·18-s − 1.14·19-s + 0.447·20-s + 0.218·21-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 2.35·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1155} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1155,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−19.47944521981931, −18.84118986011334, −18.14382745077329, −17.40267506773015, −17.15217674499403, −16.33971671531793, −15.40680978306617, −14.75176607943080, −14.10634164600144, −13.14775504443514, −12.52650706133239, −11.31069054862254, −10.77044086529249, −9.925305511860557, −9.212018498778742, −8.846741941824141, −7.870556628042501, −7.221117898961584, −6.444950830832669, −5.000306352651380, −4.127907888701044, −2.369597497356915, −1.874727272896544, 0, 1.874727272896544, 2.369597497356915, 4.127907888701044, 5.000306352651380, 6.444950830832669, 7.221117898961584, 7.870556628042501, 8.846741941824141, 9.212018498778742, 9.925305511860557, 10.77044086529249, 11.31069054862254, 12.52650706133239, 13.14775504443514, 14.10634164600144, 14.75176607943080, 15.40680978306617, 16.33971671531793, 17.15217674499403, 17.40267506773015, 18.14382745077329, 18.84118986011334, 19.47944521981931

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