Properties

Label 2-1452-1.1-c3-0-44
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 − 3.88·5-s − 1.97·7-s + 9·9-s − 48.0·13-s − 11.6·15-s − 8.31·17-s + 102.·19-s − 5.92·21-s + 108.·23-s − 109.·25-s + 27·27-s − 88.1·29-s + 71.7·31-s + 7.67·35-s + 203.·37-s − 144.·39-s − 177.·41-s − 103.·43-s − 34.9·45-s + 185.·47-s − 339.·49-s − 24.9·51-s − 698.·53-s + 306.·57-s − 157.·59-s − 207.·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.347·5-s − 0.106·7-s + 0.333·9-s − 1.02·13-s − 0.200·15-s − 0.118·17-s + 1.23·19-s − 0.0616·21-s + 0.979·23-s − 0.879·25-s + 0.192·27-s − 0.564·29-s + 0.415·31-s + 0.0370·35-s + 0.903·37-s − 0.591·39-s − 0.674·41-s − 0.367·43-s − 0.115·45-s + 0.574·47-s − 0.988·49-s − 0.0684·51-s − 1.81·53-s + 0.711·57-s − 0.347·59-s − 0.434·61-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 \( 1 + 3.88T + 125T^{2} \)
7 \( 1 + 1.97T + 343T^{2} \)
13 \( 1 + 48.0T + 2.19e3T^{2} \)
17 \( 1 + 8.31T + 4.91e3T^{2} \)
19 \( 1 - 102.T + 6.85e3T^{2} \)
23 \( 1 - 108.T + 1.21e4T^{2} \)
29 \( 1 + 88.1T + 2.43e4T^{2} \)
31 \( 1 - 71.7T + 2.97e4T^{2} \)
37 \( 1 - 203.T + 5.06e4T^{2} \)
41 \( 1 + 177.T + 6.89e4T^{2} \)
43 \( 1 + 103.T + 7.95e4T^{2} \)
47 \( 1 - 185.T + 1.03e5T^{2} \)
53 \( 1 + 698.T + 1.48e5T^{2} \)
59 \( 1 + 157.T + 2.05e5T^{2} \)
61 \( 1 + 207.T + 2.26e5T^{2} \)
67 \( 1 - 404.T + 3.00e5T^{2} \)
71 \( 1 + 22.5T + 3.57e5T^{2} \)
73 \( 1 + 848.T + 3.89e5T^{2} \)
79 \( 1 + 526.T + 4.93e5T^{2} \)
83 \( 1 + 657.T + 5.71e5T^{2} \)
89 \( 1 + 102.T + 7.04e5T^{2} \)
97 \( 1 - 321.T + 9.12e5T^{2} \)
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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

−8.799248418458487048106912500315, −7.77465731449518763249834025060, −7.41141014043911704003836230741, −6.41336648120866664409955558038, −5.28135320326455316937938211885, −4.49462929010717053625927894229, −3.41738221871753586258339994566, −2.65776229510916167669269408234, −1.41501527567056069675377497482, 0, 1.41501527567056069675377497482, 2.65776229510916167669269408234, 3.41738221871753586258339994566, 4.49462929010717053625927894229, 5.28135320326455316937938211885, 6.41336648120866664409955558038, 7.41141014043911704003836230741, 7.77465731449518763249834025060, 8.799248418458487048106912500315

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