Properties

Label 2-1452-1.1-c3-0-48
Degree $2$
Conductor $1452$
Sign $-1$
Analytic cond. $85.6707$
Root an. cond. $9.25585$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 10·5-s − 8·7-s + 9·9-s − 18·13-s + 30·15-s − 46·17-s − 40·19-s − 24·21-s + 44·23-s − 25·25-s + 27·27-s − 186·29-s − 72·31-s − 80·35-s − 114·37-s − 54·39-s − 174·41-s + 416·43-s + 90·45-s − 156·47-s − 279·49-s − 138·51-s − 62·53-s − 120·57-s − 348·59-s + 446·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.431·7-s + 1/3·9-s − 0.384·13-s + 0.516·15-s − 0.656·17-s − 0.482·19-s − 0.249·21-s + 0.398·23-s − 1/5·25-s + 0.192·27-s − 1.19·29-s − 0.417·31-s − 0.386·35-s − 0.506·37-s − 0.221·39-s − 0.662·41-s + 1.47·43-s + 0.298·45-s − 0.484·47-s − 0.813·49-s − 0.378·51-s − 0.160·53-s − 0.278·57-s − 0.767·59-s + 0.936·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(85.6707\)
Root analytic conductor: \(9.25585\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1452,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
11 \( 1 \)
good5 \( 1 - 2 p T + p^{3} T^{2} \)
7 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 + 18 T + p^{3} T^{2} \)
17 \( 1 + 46 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 - 44 T + p^{3} T^{2} \)
29 \( 1 + 186 T + p^{3} T^{2} \)
31 \( 1 + 72 T + p^{3} T^{2} \)
37 \( 1 + 114 T + p^{3} T^{2} \)
41 \( 1 + 174 T + p^{3} T^{2} \)
43 \( 1 - 416 T + p^{3} T^{2} \)
47 \( 1 + 156 T + p^{3} T^{2} \)
53 \( 1 + 62 T + p^{3} T^{2} \)
59 \( 1 + 348 T + p^{3} T^{2} \)
61 \( 1 - 446 T + p^{3} T^{2} \)
67 \( 1 + 956 T + p^{3} T^{2} \)
71 \( 1 + 444 T + p^{3} T^{2} \)
73 \( 1 + 306 T + p^{3} T^{2} \)
79 \( 1 - 664 T + p^{3} T^{2} \)
83 \( 1 - 124 T + p^{3} T^{2} \)
89 \( 1 - 602 T + p^{3} T^{2} \)
97 \( 1 - 1522 T + p^{3} T^{2} \)
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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

−9.050912323157290786836872974608, −7.966366703177066970958253311767, −7.11828434214099501808702168780, −6.32130806859449729964291023496, −5.49642607075214058764190109120, −4.47378861915352433598949584823, −3.43854538481929695406516741479, −2.42463673494848164617051597277, −1.63084883934094757699185800751, 0, 1.63084883934094757699185800751, 2.42463673494848164617051597277, 3.43854538481929695406516741479, 4.47378861915352433598949584823, 5.49642607075214058764190109120, 6.32130806859449729964291023496, 7.11828434214099501808702168780, 7.966366703177066970958253311767, 9.050912323157290786836872974608

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