Properties

Label 2-111-111.110-c0-0-1
Degree $2$
Conductor $111$
Sign $1$
Analytic cond. $0.0553962$
Root an. cond. $0.235364$
Motivic weight $0$
Arithmetic yes
Rational yes
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

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.0553962\)
Root analytic conductor: \(0.235364\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{111} (110, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5829903212\)
\(L(\frac12)\) \(\approx\) \(0.5829903212\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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