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Properties

Label 2-114444-1.1-c1-0-3
Degree $2$
Conductor $114444$
Sign $1$
Analytic cond. $913.839$
Root an. cond. $30.2297$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·7-s + 11-s − 2·13-s − 6·19-s − 25-s − 8·29-s + 8·31-s + 4·35-s − 10·37-s + 8·41-s − 2·43-s + 8·47-s − 3·49-s + 2·53-s + 2·55-s − 12·59-s − 10·61-s − 4·65-s + 12·67-s + 8·71-s − 6·73-s + 2·77-s + 2·79-s − 16·83-s + 14·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s − 1.37·19-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s + 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.269·55-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.227·77-s + 0.225·79-s − 1.75·83-s + 1.48·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114444\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(913.839\)
Root analytic conductor: \(30.2297\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 114444,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.554312198\)
\(L(\frac12)\) \(\approx\) \(2.554312198\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.90880373570972, −13.09059732647660, −12.66207776314892, −12.24851072366906, −11.63463215396161, −11.16851048885825, −10.66158736495250, −10.20805725217260, −9.751329967329972, −9.120207486706490, −8.816311955991912, −8.201085761090665, −7.614354964221998, −7.222102978733215, −6.403332859211064, −6.149758453024894, −5.527511572909291, −4.959870223238103, −4.467619717262482, −3.916951568170974, −3.168607540542174, −2.351228216934490, −1.975942932192749, −1.448061607889254, −0.4697079098342450, 0.4697079098342450, 1.448061607889254, 1.975942932192749, 2.351228216934490, 3.168607540542174, 3.916951568170974, 4.467619717262482, 4.959870223238103, 5.527511572909291, 6.149758453024894, 6.403332859211064, 7.222102978733215, 7.614354964221998, 8.201085761090665, 8.816311955991912, 9.120207486706490, 9.751329967329972, 10.20805725217260, 10.66158736495250, 11.16851048885825, 11.63463215396161, 12.24851072366906, 12.66207776314892, 13.09059732647660, 13.90880373570972

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