Properties

Label 2-42e2-1.1-c3-0-38
Degree $2$
Conductor $1764$
Sign $-1$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19·13-s + 107·19-s − 125·25-s − 289·31-s + 323·37-s + 71·43-s + 182·61-s − 127·67-s − 271·73-s − 1.38e3·79-s − 1.33e3·97-s − 1.80e3·103-s + 2.21e3·109-s + ⋯
L(s)  = 1  − 0.405·13-s + 1.29·19-s − 25-s − 1.67·31-s + 1.43·37-s + 0.251·43-s + 0.382·61-s − 0.231·67-s − 0.434·73-s − 1.97·79-s − 1.39·97-s − 1.72·103-s + 1.94·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1764} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 + 19 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 - 107 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 + 289 T + p^{3} T^{2} \)
37 \( 1 - 323 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 - 71 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 - 182 T + p^{3} T^{2} \)
67 \( 1 + 127 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 + 271 T + p^{3} T^{2} \)
79 \( 1 + 1387 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + 1330 T + p^{3} T^{2} \)
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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

−8.545843733685224632714726384741, −7.61390269976383885758839905506, −7.16288905228318448190789636938, −5.99066425361125673322565969009, −5.38094305432879129202478260369, −4.36414772300130078373076281064, −3.45137754740827241561592182715, −2.43687262764253220589504777569, −1.29628243589113686695676520543, 0, 1.29628243589113686695676520543, 2.43687262764253220589504777569, 3.45137754740827241561592182715, 4.36414772300130078373076281064, 5.38094305432879129202478260369, 5.99066425361125673322565969009, 7.16288905228318448190789636938, 7.61390269976383885758839905506, 8.545843733685224632714726384741

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