Properties

Label 2-1452-1.1-c3-0-52
Degree $2$
Conductor $1452$
Sign $-1$
Analytic cond. $85.6707$
Root an. cond. $9.25585$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 10.3·5-s + 5.80·7-s + 9·9-s − 47.3·13-s + 31.0·15-s − 109.·17-s − 11.8·19-s + 17.4·21-s − 7.30·23-s − 18.1·25-s + 27·27-s − 36.1·29-s + 117.·31-s + 59.9·35-s − 396.·37-s − 141.·39-s − 256.·41-s − 351.·43-s + 93.0·45-s − 354.·47-s − 309.·49-s − 329.·51-s − 162.·53-s − 35.5·57-s + 285.·59-s + 36.8·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.924·5-s + 0.313·7-s + 0.333·9-s − 1.00·13-s + 0.533·15-s − 1.56·17-s − 0.142·19-s + 0.180·21-s − 0.0662·23-s − 0.145·25-s + 0.192·27-s − 0.231·29-s + 0.681·31-s + 0.289·35-s − 1.76·37-s − 0.582·39-s − 0.977·41-s − 1.24·43-s + 0.308·45-s − 1.10·47-s − 0.901·49-s − 0.903·51-s − 0.421·53-s − 0.0825·57-s + 0.630·59-s + 0.0773·61-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: no
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 \( 1 - 3T \)
11 \( 1 \)
good5 \( 1 - 10.3T + 125T^{2} \)
7 \( 1 - 5.80T + 343T^{2} \)
13 \( 1 + 47.3T + 2.19e3T^{2} \)
17 \( 1 + 109.T + 4.91e3T^{2} \)
19 \( 1 + 11.8T + 6.85e3T^{2} \)
23 \( 1 + 7.30T + 1.21e4T^{2} \)
29 \( 1 + 36.1T + 2.43e4T^{2} \)
31 \( 1 - 117.T + 2.97e4T^{2} \)
37 \( 1 + 396.T + 5.06e4T^{2} \)
41 \( 1 + 256.T + 6.89e4T^{2} \)
43 \( 1 + 351.T + 7.95e4T^{2} \)
47 \( 1 + 354.T + 1.03e5T^{2} \)
53 \( 1 + 162.T + 1.48e5T^{2} \)
59 \( 1 - 285.T + 2.05e5T^{2} \)
61 \( 1 - 36.8T + 2.26e5T^{2} \)
67 \( 1 - 856.T + 3.00e5T^{2} \)
71 \( 1 - 516.T + 3.57e5T^{2} \)
73 \( 1 - 639.T + 3.89e5T^{2} \)
79 \( 1 + 613.T + 4.93e5T^{2} \)
83 \( 1 + 478.T + 5.71e5T^{2} \)
89 \( 1 + 46.9T + 7.04e5T^{2} \)
97 \( 1 - 163.T + 9.12e5T^{2} \)
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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

−8.701752026041907174976124626136, −8.157678551853258277965326300565, −6.99023310661189556995729595759, −6.49820165239523343730645328223, −5.26022797646197492811551240592, −4.64908394078915974950650330066, −3.44129221178913265019969049025, −2.28562296463619814834989540744, −1.74290141971405579050444446675, 0, 1.74290141971405579050444446675, 2.28562296463619814834989540744, 3.44129221178913265019969049025, 4.64908394078915974950650330066, 5.26022797646197492811551240592, 6.49820165239523343730645328223, 6.99023310661189556995729595759, 8.157678551853258277965326300565, 8.701752026041907174976124626136

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