Properties

Label 2-1452-1.1-c3-0-36
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 − 17.0·5-s + 13.6·7-s + 9·9-s − 11.1·13-s − 51.0·15-s + 88.8·17-s − 78.1·19-s + 41.0·21-s − 106.·23-s + 164.·25-s + 27·27-s + 233.·29-s − 243.·31-s − 232.·35-s + 368.·37-s − 33.5·39-s + 35.2·41-s − 187.·43-s − 153.·45-s − 286.·47-s − 156.·49-s + 266.·51-s − 183.·53-s − 234.·57-s + 257.·59-s + 472.·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.52·5-s + 0.738·7-s + 0.333·9-s − 0.238·13-s − 0.878·15-s + 1.26·17-s − 0.944·19-s + 0.426·21-s − 0.966·23-s + 1.31·25-s + 0.192·27-s + 1.49·29-s − 1.41·31-s − 1.12·35-s + 1.63·37-s − 0.137·39-s + 0.134·41-s − 0.665·43-s − 0.507·45-s − 0.888·47-s − 0.455·49-s + 0.732·51-s − 0.476·53-s − 0.545·57-s + 0.567·59-s + 0.992·61-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 \( 1 + 17.0T + 125T^{2} \)
7 \( 1 - 13.6T + 343T^{2} \)
13 \( 1 + 11.1T + 2.19e3T^{2} \)
17 \( 1 - 88.8T + 4.91e3T^{2} \)
19 \( 1 + 78.1T + 6.85e3T^{2} \)
23 \( 1 + 106.T + 1.21e4T^{2} \)
29 \( 1 - 233.T + 2.43e4T^{2} \)
31 \( 1 + 243.T + 2.97e4T^{2} \)
37 \( 1 - 368.T + 5.06e4T^{2} \)
41 \( 1 - 35.2T + 6.89e4T^{2} \)
43 \( 1 + 187.T + 7.95e4T^{2} \)
47 \( 1 + 286.T + 1.03e5T^{2} \)
53 \( 1 + 183.T + 1.48e5T^{2} \)
59 \( 1 - 257.T + 2.05e5T^{2} \)
61 \( 1 - 472.T + 2.26e5T^{2} \)
67 \( 1 + 138.T + 3.00e5T^{2} \)
71 \( 1 - 838.T + 3.57e5T^{2} \)
73 \( 1 + 1.14e3T + 3.89e5T^{2} \)
79 \( 1 - 203.T + 4.93e5T^{2} \)
83 \( 1 + 624.T + 5.71e5T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + 519.T + 9.12e5T^{2} \)
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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

−8.316139420773956686168361293492, −8.118499533176491484010492475015, −7.45037655999422478398263581577, −6.47462420665764133316130090175, −5.19794103235540326677959283779, −4.30042536721481508847453519324, −3.68606743948245557998579395262, −2.63414632707389927597894897753, −1.32450312035896482256139278348, 0, 1.32450312035896482256139278348, 2.63414632707389927597894897753, 3.68606743948245557998579395262, 4.30042536721481508847453519324, 5.19794103235540326677959283779, 6.47462420665764133316130090175, 7.45037655999422478398263581577, 8.118499533176491484010492475015, 8.316139420773956686168361293492

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