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  − 3-s − 2·4-s − 5-s + 7-s + 9-s + 11-s + 2·12-s + 15-s + 4·16-s + 3·17-s − 3·19-s + 2·20-s − 21-s − 23-s + 25-s − 27-s − 2·28-s − 7·29-s + 6·31-s − 33-s − 35-s − 2·36-s − 8·37-s − 2·41-s − 5·43-s − 2·44-s − 45-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.258·15-s + 16-s + 0.727·17-s − 0.688·19-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.29·29-s + 1.07·31-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 1.31·37-s − 0.312·41-s − 0.762·43-s − 0.301·44-s − 0.149·45-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  :  $\chi_{1155} (1, \cdot )$
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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
good2 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−9.430337462578200218062202779039, −8.496053516320620371043593499931, −7.899830020151268989852881692339, −6.88447028715167810681872577779, −5.84009967093002728684369364895, −5.01570492177079781617268790956, −4.24991801630843389574934371777, −3.35399164298404557991082474166, −1.49628319683318589226316651232, 0, 1.49628319683318589226316651232, 3.35399164298404557991082474166, 4.24991801630843389574934371777, 5.01570492177079781617268790956, 5.84009967093002728684369364895, 6.88447028715167810681872577779, 7.899830020151268989852881692339, 8.496053516320620371043593499931, 9.430337462578200218062202779039

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