Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.546·2-s + 3-s − 1.70·4-s − 5-s + 0.546·6-s − 7-s − 2.02·8-s + 9-s − 0.546·10-s − 11-s − 1.70·12-s − 0.568·13-s − 0.546·14-s − 15-s + 2.29·16-s + 2.70·17-s + 0.546·18-s + 6.62·19-s + 1.70·20-s − 21-s − 0.546·22-s + 8.93·23-s − 2.02·24-s + 25-s − 0.310·26-s + 27-s + 1.70·28-s + ⋯
L(s)  = 1  + 0.386·2-s + 0.577·3-s − 0.850·4-s − 0.447·5-s + 0.223·6-s − 0.377·7-s − 0.714·8-s + 0.333·9-s − 0.172·10-s − 0.301·11-s − 0.491·12-s − 0.157·13-s − 0.146·14-s − 0.258·15-s + 0.574·16-s + 0.655·17-s + 0.128·18-s + 1.51·19-s + 0.380·20-s − 0.218·21-s − 0.116·22-s + 1.86·23-s − 0.412·24-s + 0.200·25-s − 0.0609·26-s + 0.192·27-s + 0.321·28-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  =  \(0\)
Selberg data  =  \((2,\ 1155,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.740671324\)
\(L(\frac12)\)  \(\approx\)  \(1.740671324\)
\(L(\frac{3}{2})\)   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 \( 1 - 0.546T + 2T^{2} \)
13 \( 1 + 0.568T + 13T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 - 6.62T + 19T^{2} \)
23 \( 1 - 8.93T + 23T^{2} \)
29 \( 1 - 5.27T + 29T^{2} \)
31 \( 1 - 9.66T + 31T^{2} \)
37 \( 1 + 2.75T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 0.0520T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 - 6.36T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 - 2.56T + 71T^{2} \)
73 \( 1 - 2.26T + 73T^{2} \)
79 \( 1 + 2.56T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−9.788760164612552487018430933491, −8.909300447377749185304620101622, −8.265538531157894107758249806283, −7.42122693331536317880620756398, −6.48237852303071604337493604741, −5.19062424828374817329144499381, −4.71013548150215139994112787272, −3.36591280884191110978989785576, −3.03120596655070389442266982397, −0.955862374071830574861273941973, 0.955862374071830574861273941973, 3.03120596655070389442266982397, 3.36591280884191110978989785576, 4.71013548150215139994112787272, 5.19062424828374817329144499381, 6.48237852303071604337493604741, 7.42122693331536317880620756398, 8.265538531157894107758249806283, 8.909300447377749185304620101622, 9.788760164612552487018430933491

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