Properties

Label 2-42e2-1.1-c3-0-40
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  + 6·5-s − 36·11-s − 62·13-s + 114·17-s + 76·19-s + 24·23-s − 89·25-s − 54·29-s + 112·31-s − 178·37-s + 378·41-s − 172·43-s − 192·47-s + 402·53-s − 216·55-s + 396·59-s − 254·61-s − 372·65-s − 1.01e3·67-s − 840·71-s − 890·73-s + 80·79-s − 108·83-s + 684·85-s − 1.63e3·89-s + 456·95-s − 1.01e3·97-s + ⋯
L(s)  = 1  + 0.536·5-s − 0.986·11-s − 1.32·13-s + 1.62·17-s + 0.917·19-s + 0.217·23-s − 0.711·25-s − 0.345·29-s + 0.648·31-s − 0.790·37-s + 1.43·41-s − 0.609·43-s − 0.595·47-s + 1.04·53-s − 0.529·55-s + 0.873·59-s − 0.533·61-s − 0.709·65-s − 1.84·67-s − 1.40·71-s − 1.42·73-s + 0.113·79-s − 0.142·83-s + 0.872·85-s − 1.95·89-s + 0.492·95-s − 1.05·97-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 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 + 62 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
19 \( 1 - 4 p T + p^{3} T^{2} \)
23 \( 1 - 24 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 - 112 T + p^{3} T^{2} \)
37 \( 1 + 178 T + p^{3} T^{2} \)
41 \( 1 - 378 T + p^{3} T^{2} \)
43 \( 1 + 4 p T + p^{3} T^{2} \)
47 \( 1 + 192 T + p^{3} T^{2} \)
53 \( 1 - 402 T + p^{3} T^{2} \)
59 \( 1 - 396 T + p^{3} T^{2} \)
61 \( 1 + 254 T + p^{3} T^{2} \)
67 \( 1 + 1012 T + p^{3} T^{2} \)
71 \( 1 + 840 T + p^{3} T^{2} \)
73 \( 1 + 890 T + p^{3} T^{2} \)
79 \( 1 - 80 T + p^{3} T^{2} \)
83 \( 1 + 108 T + p^{3} T^{2} \)
89 \( 1 + 1638 T + p^{3} T^{2} \)
97 \( 1 + 1010 T + p^{3} T^{2} \)
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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

−8.494894190072557628145194209147, −7.56684478510705097588889290301, −7.24736972999837389283448874484, −5.83832557486517577753966928362, −5.43435622037117527746545667685, −4.54802249390616255448756347006, −3.23596547778843886238172308840, −2.51005111375962905442368453532, −1.32979962514739097168535687556, 0, 1.32979962514739097168535687556, 2.51005111375962905442368453532, 3.23596547778843886238172308840, 4.54802249390616255448756347006, 5.43435622037117527746545667685, 5.83832557486517577753966928362, 7.24736972999837389283448874484, 7.56684478510705097588889290301, 8.494894190072557628145194209147

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