Properties

Label 2-42e2-1.1-c3-0-42
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  + 4·5-s + 20·11-s + 4·13-s + 24·17-s − 44·19-s − 72·23-s − 109·25-s + 38·29-s − 184·31-s − 30·37-s − 216·41-s − 164·43-s + 520·47-s + 146·53-s + 80·55-s + 460·59-s − 628·61-s + 16·65-s + 556·67-s − 592·71-s − 1.02e3·73-s − 104·79-s − 324·83-s + 96·85-s + 896·89-s − 176·95-s + 920·97-s + ⋯
L(s)  = 1  + 0.357·5-s + 0.548·11-s + 0.0853·13-s + 0.342·17-s − 0.531·19-s − 0.652·23-s − 0.871·25-s + 0.243·29-s − 1.06·31-s − 0.133·37-s − 0.822·41-s − 0.581·43-s + 1.61·47-s + 0.378·53-s + 0.196·55-s + 1.01·59-s − 1.31·61-s + 0.0305·65-s + 1.01·67-s − 0.989·71-s − 1.64·73-s − 0.148·79-s − 0.428·83-s + 0.122·85-s + 1.06·89-s − 0.190·95-s + 0.963·97-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 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 - 20 T + p^{3} T^{2} \)
13 \( 1 - 4 T + p^{3} T^{2} \)
17 \( 1 - 24 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 - 38 T + p^{3} T^{2} \)
31 \( 1 + 184 T + p^{3} T^{2} \)
37 \( 1 + 30 T + p^{3} T^{2} \)
41 \( 1 + 216 T + p^{3} T^{2} \)
43 \( 1 + 164 T + p^{3} T^{2} \)
47 \( 1 - 520 T + p^{3} T^{2} \)
53 \( 1 - 146 T + p^{3} T^{2} \)
59 \( 1 - 460 T + p^{3} T^{2} \)
61 \( 1 + 628 T + p^{3} T^{2} \)
67 \( 1 - 556 T + p^{3} T^{2} \)
71 \( 1 + 592 T + p^{3} T^{2} \)
73 \( 1 + 1024 T + p^{3} T^{2} \)
79 \( 1 + 104 T + p^{3} T^{2} \)
83 \( 1 + 324 T + p^{3} T^{2} \)
89 \( 1 - 896 T + p^{3} T^{2} \)
97 \( 1 - 920 T + p^{3} T^{2} \)
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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

−8.631824740861872057556416940388, −7.75184053469148421510111225678, −6.92805008892479848421258759118, −6.07581117431351301630670690230, −5.41949830623708835622798278344, −4.29470056156461000616520711169, −3.53768921504853255726080105287, −2.31314883671426183931285282433, −1.39407604970780340297145861651, 0, 1.39407604970780340297145861651, 2.31314883671426183931285282433, 3.53768921504853255726080105287, 4.29470056156461000616520711169, 5.41949830623708835622798278344, 6.07581117431351301630670690230, 6.92805008892479848421258759118, 7.75184053469148421510111225678, 8.631824740861872057556416940388

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