Properties

Label 2-1452-1.1-c3-0-43
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 + 3.10·5-s − 29.8·7-s + 9·9-s + 32.2·13-s + 9.32·15-s + 70.3·17-s − 47.1·19-s − 89.5·21-s + 56.3·23-s − 115.·25-s + 27·27-s − 194.·29-s + 146.·31-s − 92.8·35-s + 310.·37-s + 96.6·39-s − 332.·41-s + 87.1·43-s + 27.9·45-s − 101.·47-s + 548.·49-s + 211.·51-s + 365.·53-s − 141.·57-s − 641.·59-s − 758.·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.278·5-s − 1.61·7-s + 0.333·9-s + 0.687·13-s + 0.160·15-s + 1.00·17-s − 0.569·19-s − 0.930·21-s + 0.510·23-s − 0.922·25-s + 0.192·27-s − 1.24·29-s + 0.847·31-s − 0.448·35-s + 1.37·37-s + 0.396·39-s − 1.26·41-s + 0.309·43-s + 0.0926·45-s − 0.313·47-s + 1.59·49-s + 0.579·51-s + 0.948·53-s − 0.329·57-s − 1.41·59-s − 1.59·61-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 \( 1 - 3.10T + 125T^{2} \)
7 \( 1 + 29.8T + 343T^{2} \)
13 \( 1 - 32.2T + 2.19e3T^{2} \)
17 \( 1 - 70.3T + 4.91e3T^{2} \)
19 \( 1 + 47.1T + 6.85e3T^{2} \)
23 \( 1 - 56.3T + 1.21e4T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 - 146.T + 2.97e4T^{2} \)
37 \( 1 - 310.T + 5.06e4T^{2} \)
41 \( 1 + 332.T + 6.89e4T^{2} \)
43 \( 1 - 87.1T + 7.95e4T^{2} \)
47 \( 1 + 101.T + 1.03e5T^{2} \)
53 \( 1 - 365.T + 1.48e5T^{2} \)
59 \( 1 + 641.T + 2.05e5T^{2} \)
61 \( 1 + 758.T + 2.26e5T^{2} \)
67 \( 1 - 123.T + 3.00e5T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 - 525.T + 3.89e5T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + 330.T + 5.71e5T^{2} \)
89 \( 1 + 300.T + 7.04e5T^{2} \)
97 \( 1 + 1.54e3T + 9.12e5T^{2} \)
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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

−8.891313697675237811208944925312, −7.979911532692285935535455626738, −7.12552213815139054670669086242, −6.24106499013750700561470229954, −5.69503431258709705368858630777, −4.25893390783284540291982143565, −3.41320644799502148767569716668, −2.72393088161907501226091609733, −1.40426013326333939147657147673, 0, 1.40426013326333939147657147673, 2.72393088161907501226091609733, 3.41320644799502148767569716668, 4.25893390783284540291982143565, 5.69503431258709705368858630777, 6.24106499013750700561470229954, 7.12552213815139054670669086242, 7.979911532692285935535455626738, 8.891313697675237811208944925312

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