Properties

Label 2-1452-1.1-c3-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·7-s + 9·9-s + 88·13-s + 66·17-s + 40·19-s + 6·21-s + 6·23-s − 125·25-s − 27·27-s + 54·29-s + 8·31-s − 106·37-s − 264·39-s − 354·41-s + 124·43-s + 546·47-s − 339·49-s − 198·51-s − 408·53-s − 120·57-s + 552·59-s − 404·61-s − 18·63-s − 4·67-s − 18·69-s + 126·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.107·7-s + 1/3·9-s + 1.87·13-s + 0.941·17-s + 0.482·19-s + 0.0623·21-s + 0.0543·23-s − 25-s − 0.192·27-s + 0.345·29-s + 0.0463·31-s − 0.470·37-s − 1.08·39-s − 1.34·41-s + 0.439·43-s + 1.69·47-s − 0.988·49-s − 0.543·51-s − 1.05·53-s − 0.278·57-s + 1.21·59-s − 0.847·61-s − 0.0359·63-s − 0.00729·67-s − 0.0314·69-s + 0.210·71-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: \(0\)
Selberg data: \((2,\ 1452,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.932776477\)
\(L(\frac12)\) \(\approx\) \(1.932776477\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
13 \( 1 - 88 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 - 6 T + p^{3} T^{2} \)
29 \( 1 - 54 T + p^{3} T^{2} \)
31 \( 1 - 8 T + p^{3} T^{2} \)
37 \( 1 + 106 T + p^{3} T^{2} \)
41 \( 1 + 354 T + p^{3} T^{2} \)
43 \( 1 - 124 T + p^{3} T^{2} \)
47 \( 1 - 546 T + p^{3} T^{2} \)
53 \( 1 + 408 T + p^{3} T^{2} \)
59 \( 1 - 552 T + p^{3} T^{2} \)
61 \( 1 + 404 T + p^{3} T^{2} \)
67 \( 1 + 4 T + p^{3} T^{2} \)
71 \( 1 - 126 T + p^{3} T^{2} \)
73 \( 1 - 166 T + p^{3} T^{2} \)
79 \( 1 - 874 T + p^{3} T^{2} \)
83 \( 1 + 444 T + p^{3} T^{2} \)
89 \( 1 - 1002 T + p^{3} T^{2} \)
97 \( 1 + 802 T + p^{3} T^{2} \)
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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

−9.173884959032109857617080866918, −8.320775897865310082610962034545, −7.56262676829814614117345571647, −6.52616094370776621721910377448, −5.90724123445452207683662210334, −5.14023343120059289357325272482, −3.95457127920995794386684814602, −3.25619103014024700050532247527, −1.69588811526127725744378270964, −0.74491128665918822880351383950, 0.74491128665918822880351383950, 1.69588811526127725744378270964, 3.25619103014024700050532247527, 3.95457127920995794386684814602, 5.14023343120059289357325272482, 5.90724123445452207683662210334, 6.52616094370776621721910377448, 7.56262676829814614117345571647, 8.320775897865310082610962034545, 9.173884959032109857617080866918

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