Properties

Degree 2
Conductor $ 3 \cdot 37 $
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 − 4-s − 2·7-s + 9-s − 12-s + 16-s − 2·21-s − 25-s + 27-s + 2·28-s − 36-s + 37-s + 48-s + 3·49-s − 2·63-s − 64-s − 2·67-s − 2·73-s − 75-s + 81-s + 2·84-s + 100-s − 108-s + 111-s − 2·112-s + ⋯
L(s)  = 1  + 3-s − 4-s − 2·7-s + 9-s − 12-s + 16-s − 2·21-s − 25-s + 27-s + 2·28-s − 36-s + 37-s + 48-s + 3·49-s − 2·63-s − 64-s − 2·67-s − 2·73-s − 75-s + 81-s + 2·84-s + 100-s − 108-s + 111-s − 2·112-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(111\)    =    \(3 \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{111} (110, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 111,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.5829903212$
$L(\frac12)$  $\approx$  $0.5829903212$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−13.61912087983946106403182530122, −13.17811219851589701555407411251, −12.30100264888277957817024823027, −10.18777066568050913245572485824, −9.568599253291209053315267014723, −8.774765372840556032518468776584, −7.44584650258568254295138388593, −6.05257527874793912859985097248, −4.14852411809580231378989051158, −3.07636964433466202517570300634, 3.07636964433466202517570300634, 4.14852411809580231378989051158, 6.05257527874793912859985097248, 7.44584650258568254295138388593, 8.774765372840556032518468776584, 9.568599253291209053315267014723, 10.18777066568050913245572485824, 12.30100264888277957817024823027, 13.17811219851589701555407411251, 13.61912087983946106403182530122

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