Properties

Label 2-42e2-1.1-c1-0-13
Degree $2$
Conductor $1764$
Sign $-1$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·13-s + 19-s − 5·25-s − 11·31-s + 11·37-s − 13·43-s − 14·61-s + 5·67-s − 17·73-s + 17·79-s − 14·97-s + 13·103-s − 19·109-s + ⋯
L(s)  = 1  − 1.38·13-s + 0.229·19-s − 25-s − 1.97·31-s + 1.80·37-s − 1.98·43-s − 1.79·61-s + 0.610·67-s − 1.98·73-s + 1.91·79-s − 1.42·97-s + 1.28·103-s − 1.81·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1764} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.072445151185707267203043951256, −7.919824911977054215538992032791, −7.46472304636129055466154416467, −6.54001911877925024909239165864, −5.58966265767998501281876191722, −4.83519218317223951659185514643, −3.86532544874840477233066388199, −2.79032016914030089273128181768, −1.74596868449437337311970648766, 0, 1.74596868449437337311970648766, 2.79032016914030089273128181768, 3.86532544874840477233066388199, 4.83519218317223951659185514643, 5.58966265767998501281876191722, 6.54001911877925024909239165864, 7.46472304636129055466154416467, 7.919824911977054215538992032791, 9.072445151185707267203043951256

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