Properties

Label 2-42e2-1.1-c3-0-29
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  − 18·5-s − 36·11-s + 10·13-s + 18·17-s + 100·19-s − 72·23-s + 199·25-s + 234·29-s + 16·31-s − 226·37-s + 90·41-s + 452·43-s + 432·47-s − 414·53-s + 648·55-s − 684·59-s − 422·61-s − 180·65-s + 332·67-s + 360·71-s − 26·73-s + 512·79-s − 1.18e3·83-s − 324·85-s − 630·89-s − 1.80e3·95-s + 1.05e3·97-s + ⋯
L(s)  = 1  − 1.60·5-s − 0.986·11-s + 0.213·13-s + 0.256·17-s + 1.20·19-s − 0.652·23-s + 1.59·25-s + 1.49·29-s + 0.0926·31-s − 1.00·37-s + 0.342·41-s + 1.60·43-s + 1.34·47-s − 1.07·53-s + 1.58·55-s − 1.50·59-s − 0.885·61-s − 0.343·65-s + 0.605·67-s + 0.601·71-s − 0.0416·73-s + 0.729·79-s − 1.57·83-s − 0.413·85-s − 0.750·89-s − 1.94·95-s + 1.10·97-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 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 - 10 T + p^{3} T^{2} \)
17 \( 1 - 18 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 - 234 T + p^{3} T^{2} \)
31 \( 1 - 16 T + p^{3} T^{2} \)
37 \( 1 + 226 T + p^{3} T^{2} \)
41 \( 1 - 90 T + p^{3} T^{2} \)
43 \( 1 - 452 T + p^{3} T^{2} \)
47 \( 1 - 432 T + p^{3} T^{2} \)
53 \( 1 + 414 T + p^{3} T^{2} \)
59 \( 1 + 684 T + p^{3} T^{2} \)
61 \( 1 + 422 T + p^{3} T^{2} \)
67 \( 1 - 332 T + p^{3} T^{2} \)
71 \( 1 - 360 T + p^{3} T^{2} \)
73 \( 1 + 26 T + p^{3} T^{2} \)
79 \( 1 - 512 T + p^{3} T^{2} \)
83 \( 1 + 1188 T + p^{3} T^{2} \)
89 \( 1 + 630 T + p^{3} T^{2} \)
97 \( 1 - 1054 T + p^{3} T^{2} \)
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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

−8.337617908676463810725599056245, −7.72277839900436686486903202696, −7.28651124892840429238187796598, −6.13749555685334373095846121630, −5.12853804731705798215557659125, −4.34692342390994105346955289328, −3.46010965783694822408001880981, −2.67721558033884666827525332055, −1.03091077808027250966867992298, 0, 1.03091077808027250966867992298, 2.67721558033884666827525332055, 3.46010965783694822408001880981, 4.34692342390994105346955289328, 5.12853804731705798215557659125, 6.13749555685334373095846121630, 7.28651124892840429238187796598, 7.72277839900436686486903202696, 8.337617908676463810725599056245

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