Properties

Label 2-42e2-1.1-c1-0-16
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  + 2·5-s − 2·11-s − 3·13-s − 8·17-s − 19-s − 8·23-s − 25-s − 4·29-s + 3·31-s − 37-s − 6·41-s + 11·43-s − 6·47-s + 12·53-s − 4·55-s − 4·59-s − 6·61-s − 6·65-s + 13·67-s + 10·71-s − 11·73-s − 3·79-s − 2·83-s − 16·85-s − 2·95-s + 10·97-s − 10·101-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.603·11-s − 0.832·13-s − 1.94·17-s − 0.229·19-s − 1.66·23-s − 1/5·25-s − 0.742·29-s + 0.538·31-s − 0.164·37-s − 0.937·41-s + 1.67·43-s − 0.875·47-s + 1.64·53-s − 0.539·55-s − 0.520·59-s − 0.768·61-s − 0.744·65-s + 1.58·67-s + 1.18·71-s − 1.28·73-s − 0.337·79-s − 0.219·83-s − 1.73·85-s − 0.205·95-s + 1.01·97-s − 0.995·101-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.979607423699584960965065753425, −8.165903023868689901440742830487, −7.28201278848798770129892287430, −6.41755506253781280834932103733, −5.72727663875301224953763371693, −4.81907659847668392336000204438, −3.96515840830956569482328085725, −2.48058544646181868251053154060, −1.98748572081014361751177818079, 0, 1.98748572081014361751177818079, 2.48058544646181868251053154060, 3.96515840830956569482328085725, 4.81907659847668392336000204438, 5.72727663875301224953763371693, 6.41755506253781280834932103733, 7.28201278848798770129892287430, 8.165903023868689901440742830487, 8.979607423699584960965065753425

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