Properties

Label 2-42e2-1.1-c3-0-35
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  − 8·5-s + 40·11-s + 12·13-s − 58·17-s − 26·19-s + 64·23-s − 61·25-s + 62·29-s − 252·31-s + 26·37-s + 6·41-s + 416·43-s − 396·47-s + 450·53-s − 320·55-s + 274·59-s + 576·61-s − 96·65-s − 476·67-s + 448·71-s + 158·73-s − 936·79-s + 530·83-s + 464·85-s − 390·89-s + 208·95-s − 214·97-s + ⋯
L(s)  = 1  − 0.715·5-s + 1.09·11-s + 0.256·13-s − 0.827·17-s − 0.313·19-s + 0.580·23-s − 0.487·25-s + 0.397·29-s − 1.46·31-s + 0.115·37-s + 0.0228·41-s + 1.47·43-s − 1.22·47-s + 1.16·53-s − 0.784·55-s + 0.604·59-s + 1.20·61-s − 0.183·65-s − 0.867·67-s + 0.748·71-s + 0.253·73-s − 1.33·79-s + 0.700·83-s + 0.592·85-s − 0.464·89-s + 0.224·95-s − 0.224·97-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 \( 1 + 8 T + p^{3} T^{2} \)
11 \( 1 - 40 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 + 58 T + p^{3} T^{2} \)
19 \( 1 + 26 T + p^{3} T^{2} \)
23 \( 1 - 64 T + p^{3} T^{2} \)
29 \( 1 - 62 T + p^{3} T^{2} \)
31 \( 1 + 252 T + p^{3} T^{2} \)
37 \( 1 - 26 T + p^{3} T^{2} \)
41 \( 1 - 6 T + p^{3} T^{2} \)
43 \( 1 - 416 T + p^{3} T^{2} \)
47 \( 1 + 396 T + p^{3} T^{2} \)
53 \( 1 - 450 T + p^{3} T^{2} \)
59 \( 1 - 274 T + p^{3} T^{2} \)
61 \( 1 - 576 T + p^{3} T^{2} \)
67 \( 1 + 476 T + p^{3} T^{2} \)
71 \( 1 - 448 T + p^{3} T^{2} \)
73 \( 1 - 158 T + p^{3} T^{2} \)
79 \( 1 + 936 T + p^{3} T^{2} \)
83 \( 1 - 530 T + p^{3} T^{2} \)
89 \( 1 + 390 T + p^{3} T^{2} \)
97 \( 1 + 214 T + p^{3} T^{2} \)
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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

−8.675960588816124773674598466039, −7.72033203803121628455544297174, −6.95935021801911658606804195124, −6.26608021194284667904602827366, −5.23627403119803726848106405498, −4.15017158593335574949758367187, −3.70990277000385297676983670531, −2.41832047346826998446542398859, −1.22857019588297415942999497643, 0, 1.22857019588297415942999497643, 2.41832047346826998446542398859, 3.70990277000385297676983670531, 4.15017158593335574949758367187, 5.23627403119803726848106405498, 6.26608021194284667904602827366, 6.95935021801911658606804195124, 7.72033203803121628455544297174, 8.675960588816124773674598466039

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