Properties

Label 2-334620-1.1-c1-0-41
Degree $2$
Conductor $334620$
Sign $-1$
Analytic cond. $2671.95$
Root an. cond. $51.6909$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 11-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 10·29-s + 4·35-s − 2·37-s − 4·41-s − 8·43-s + 8·47-s + 9·49-s + 12·53-s − 55-s − 4·59-s + 6·61-s − 10·67-s + 8·71-s − 2·73-s − 4·77-s − 10·79-s − 2·83-s − 2·85-s + 10·89-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.301·11-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.676·35-s − 0.328·37-s − 0.624·41-s − 1.21·43-s + 1.16·47-s + 9/7·49-s + 1.64·53-s − 0.134·55-s − 0.520·59-s + 0.768·61-s − 1.22·67-s + 0.949·71-s − 0.234·73-s − 0.455·77-s − 1.12·79-s − 0.219·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(334620\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(2671.95\)
Root analytic conductor: \(51.6909\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 334620,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.79585559538261, −12.15734219568298, −11.91628633235103, −11.82806161371433, −10.91827693563303, −10.40369762286356, −10.02204536737996, −9.802598282313511, −9.152461891989004, −8.724591749424689, −8.271416547562721, −7.724178397699464, −7.170909273341396, −6.747702848671069, −6.441390211703068, −5.752368338535585, −5.452006018158580, −4.713392484646942, −4.169714632240155, −3.654865487444040, −3.191960986926745, −2.825589832197093, −2.135025542764265, −1.257001822320879, −0.7088708256073762, 0, 0.7088708256073762, 1.257001822320879, 2.135025542764265, 2.825589832197093, 3.191960986926745, 3.654865487444040, 4.169714632240155, 4.713392484646942, 5.452006018158580, 5.752368338535585, 6.441390211703068, 6.747702848671069, 7.170909273341396, 7.724178397699464, 8.271416547562721, 8.724591749424689, 9.152461891989004, 9.802598282313511, 10.02204536737996, 10.40369762286356, 10.91827693563303, 11.82806161371433, 11.91628633235103, 12.15734219568298, 12.79585559538261

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