Properties

Label 2-42e2-1.1-c1-0-12
Degree $2$
Conductor $1764$
Sign $-1$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·5-s − 4·11-s + 4.24·13-s − 1.41·17-s + 2.82·19-s + 4·23-s − 2.99·25-s − 8·29-s − 8·37-s + 7.07·41-s − 4·43-s − 5.65·47-s − 10·53-s + 5.65·55-s − 14.1·59-s − 7.07·61-s − 6·65-s − 7.07·73-s + 8·79-s + 14.1·83-s + 2.00·85-s − 7.07·89-s − 4.00·95-s − 1.41·97-s + 12.7·101-s − 11.3·103-s − 8·107-s + ⋯
L(s)  = 1  − 0.632·5-s − 1.20·11-s + 1.17·13-s − 0.342·17-s + 0.648·19-s + 0.834·23-s − 0.599·25-s − 1.48·29-s − 1.31·37-s + 1.10·41-s − 0.609·43-s − 0.825·47-s − 1.37·53-s + 0.762·55-s − 1.84·59-s − 0.905·61-s − 0.744·65-s − 0.827·73-s + 0.900·79-s + 1.55·83-s + 0.216·85-s − 0.749·89-s − 0.410·95-s − 0.143·97-s + 1.26·101-s − 1.11·103-s − 0.773·107-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7.07T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + 1.41T + 97T^{2} \)
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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

−8.898965523553725324347540429630, −7.942348626848757998154827664851, −7.56913319470644501301123893089, −6.51056749404096444344159530750, −5.60569199964181428631597703535, −4.81710160233842412161692691868, −3.74058163895908899107213928068, −3.00644705922845396256935840497, −1.62213676513059815478849376922, 0, 1.62213676513059815478849376922, 3.00644705922845396256935840497, 3.74058163895908899107213928068, 4.81710160233842412161692691868, 5.60569199964181428631597703535, 6.51056749404096444344159530750, 7.56913319470644501301123893089, 7.942348626848757998154827664851, 8.898965523553725324347540429630

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