Properties

Label 2-42e2-1.1-c3-0-37
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

−8.395318092141434286079078545846, −7.88748097611709030713013954050, −6.90033205804127419173351591797, −6.23304016795959857746125527686, −5.24347821224779889180813510747, −4.30920720081669948084468200516, −3.54590827379578791265060217990, −2.43146317426484584920885845296, −1.26090442782656232916530264195, 0, 1.26090442782656232916530264195, 2.43146317426484584920885845296, 3.54590827379578791265060217990, 4.30920720081669948084468200516, 5.24347821224779889180813510747, 6.23304016795959857746125527686, 6.90033205804127419173351591797, 7.88748097611709030713013954050, 8.395318092141434286079078545846

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