Properties

Degree $2$
Conductor $111573$
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-s − 4-s − 3·8-s − 11-s − 2·13-s − 16-s − 2·19-s − 22-s − 23-s − 5·25-s − 2·26-s + 10·29-s − 4·31-s + 5·32-s + 2·37-s − 2·38-s − 2·41-s + 2·43-s + 44-s − 46-s − 8·47-s − 5·50-s + 2·52-s + 4·53-s + 10·58-s − 12·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.301·11-s − 0.554·13-s − 1/4·16-s − 0.458·19-s − 0.213·22-s − 0.208·23-s − 25-s − 0.392·26-s + 1.85·29-s − 0.718·31-s + 0.883·32-s + 0.328·37-s − 0.324·38-s − 0.312·41-s + 0.304·43-s + 0.150·44-s − 0.147·46-s − 1.16·47-s − 0.707·50-s + 0.277·52-s + 0.549·53-s + 1.31·58-s − 1.56·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111573\)    =    \(3^{2} \cdot 7^{2} \cdot 11 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{111573} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 111573,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.79836478023179, −13.55513085230400, −12.88274064843290, −12.46384861127528, −12.16006347071505, −11.60165448301267, −11.03818877773536, −10.46312574711297, −9.793344955352390, −9.678569967760953, −8.926449264166858, −8.408245682157314, −8.019214281044085, −7.432779874716931, −6.646468267579371, −6.347592263085378, −5.605743087010060, −5.272609196405523, −4.554356785034704, −4.311811631917039, −3.576544217176841, −3.027360101238752, −2.439351766440504, −1.735057148925249, −0.7274502607793759, 0, 0.7274502607793759, 1.735057148925249, 2.439351766440504, 3.027360101238752, 3.576544217176841, 4.311811631917039, 4.554356785034704, 5.272609196405523, 5.605743087010060, 6.347592263085378, 6.646468267579371, 7.432779874716931, 8.019214281044085, 8.408245682157314, 8.926449264166858, 9.678569967760953, 9.793344955352390, 10.46312574711297, 11.03818877773536, 11.60165448301267, 12.16006347071505, 12.46384861127528, 12.88274064843290, 13.55513085230400, 13.79836478023179

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