Properties

Label 2-42e2-1.1-c3-0-44
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 + 12·11-s + 82·13-s − 30·17-s − 68·19-s − 216·23-s − 89·25-s − 246·29-s + 112·31-s + 110·37-s − 246·41-s − 172·43-s + 192·47-s − 558·53-s + 72·55-s + 540·59-s − 110·61-s + 492·65-s + 140·67-s + 840·71-s + 550·73-s − 208·79-s + 516·83-s − 180·85-s − 1.39e3·89-s − 408·95-s − 1.58e3·97-s + ⋯
L(s)  = 1  + 0.536·5-s + 0.328·11-s + 1.74·13-s − 0.428·17-s − 0.821·19-s − 1.95·23-s − 0.711·25-s − 1.57·29-s + 0.648·31-s + 0.488·37-s − 0.937·41-s − 0.609·43-s + 0.595·47-s − 1.44·53-s + 0.176·55-s + 1.19·59-s − 0.230·61-s + 0.938·65-s + 0.255·67-s + 1.40·71-s + 0.881·73-s − 0.296·79-s + 0.682·83-s − 0.229·85-s − 1.66·89-s − 0.440·95-s − 1.66·97-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 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 - 82 T + p^{3} T^{2} \)
17 \( 1 + 30 T + p^{3} T^{2} \)
19 \( 1 + 68 T + p^{3} T^{2} \)
23 \( 1 + 216 T + p^{3} T^{2} \)
29 \( 1 + 246 T + p^{3} T^{2} \)
31 \( 1 - 112 T + p^{3} T^{2} \)
37 \( 1 - 110 T + p^{3} T^{2} \)
41 \( 1 + 6 p T + p^{3} T^{2} \)
43 \( 1 + 4 p T + p^{3} T^{2} \)
47 \( 1 - 192 T + p^{3} T^{2} \)
53 \( 1 + 558 T + p^{3} T^{2} \)
59 \( 1 - 540 T + p^{3} T^{2} \)
61 \( 1 + 110 T + p^{3} T^{2} \)
67 \( 1 - 140 T + p^{3} T^{2} \)
71 \( 1 - 840 T + p^{3} T^{2} \)
73 \( 1 - 550 T + p^{3} T^{2} \)
79 \( 1 + 208 T + p^{3} T^{2} \)
83 \( 1 - 516 T + p^{3} T^{2} \)
89 \( 1 + 1398 T + p^{3} T^{2} \)
97 \( 1 + 1586 T + p^{3} T^{2} \)
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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

−8.438762494670362863964694086153, −8.007065024776906709255659445450, −6.69983176620356646016116611703, −6.15722172992460253299256757843, −5.49826090771971409946405709650, −4.16189100294200809516396173417, −3.65962886513834044890905794259, −2.21734621807336752729527798061, −1.46518633748361488360563753889, 0, 1.46518633748361488360563753889, 2.21734621807336752729527798061, 3.65962886513834044890905794259, 4.16189100294200809516396173417, 5.49826090771971409946405709650, 6.15722172992460253299256757843, 6.69983176620356646016116611703, 8.007065024776906709255659445450, 8.438762494670362863964694086153

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