Properties

Label 2-42e2-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 3·11-s − 2·13-s + 3·17-s + 19-s − 3·23-s + 4·25-s + 6·29-s + 7·31-s − 37-s + 6·41-s − 4·43-s − 9·47-s − 3·53-s + 9·55-s + 9·59-s + 61-s − 6·65-s − 7·67-s + 73-s − 13·79-s + 12·83-s + 9·85-s + 15·89-s + 3·95-s + 10·97-s + 15·101-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.904·11-s − 0.554·13-s + 0.727·17-s + 0.229·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s + 1.25·31-s − 0.164·37-s + 0.937·41-s − 0.609·43-s − 1.31·47-s − 0.412·53-s + 1.21·55-s + 1.17·59-s + 0.128·61-s − 0.744·65-s − 0.855·67-s + 0.117·73-s − 1.46·79-s + 1.31·83-s + 0.976·85-s + 1.58·89-s + 0.307·95-s + 1.01·97-s + 1.49·101-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: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.394052064\)
\(L(\frac12)\) \(\approx\) \(2.394052064\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−9.511672581695242399587565478770, −8.608788850096275615758357415082, −7.75445996130784883001003386200, −6.66804808324987664685271338563, −6.17371737327401791885640284583, −5.30404673676025296724252237112, −4.44653635446383902080069146788, −3.21812548500746171890057674885, −2.20240849637471555115239450077, −1.16213428834184120958012944154, 1.16213428834184120958012944154, 2.20240849637471555115239450077, 3.21812548500746171890057674885, 4.44653635446383902080069146788, 5.30404673676025296724252237112, 6.17371737327401791885640284583, 6.66804808324987664685271338563, 7.75445996130784883001003386200, 8.608788850096275615758357415082, 9.511672581695242399587565478770

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