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Properties

Label 2-29400-1.1-c1-0-60
Degree $2$
Conductor $29400$
Sign $-1$
Analytic cond. $234.760$
Root an. cond. $15.3218$
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  − 3-s + 9-s − 3·11-s − 2·13-s − 2·19-s − 7·23-s − 27-s − 3·29-s + 6·31-s + 3·33-s + 3·37-s + 2·39-s − 5·43-s + 2·47-s − 2·53-s + 2·57-s + 10·59-s + 8·61-s + 9·67-s + 7·69-s + 9·71-s − 8·73-s − 79-s + 81-s + 14·83-s + 3·87-s + 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 0.458·19-s − 1.45·23-s − 0.192·27-s − 0.557·29-s + 1.07·31-s + 0.522·33-s + 0.493·37-s + 0.320·39-s − 0.762·43-s + 0.291·47-s − 0.274·53-s + 0.264·57-s + 1.30·59-s + 1.02·61-s + 1.09·67-s + 0.842·69-s + 1.06·71-s − 0.936·73-s − 0.112·79-s + 1/9·81-s + 1.53·83-s + 0.321·87-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(234.760\)
Root analytic conductor: \(15.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.43626471374847, −14.98960377008557, −14.39231318351811, −13.77682373761716, −13.25325350561011, −12.71773018306505, −12.26832834445547, −11.62611689580867, −11.26690806255471, −10.50532650248705, −10.04618649388703, −9.756087280135277, −8.878739252166391, −8.173443943169182, −7.829288998111871, −7.128719533342675, −6.482960837011903, −5.957404054878986, −5.294226680249339, −4.804701969809159, −4.106570214901189, −3.439752886203170, −2.426317196740967, −2.039758592881155, −0.8393211460915303, 0, 0.8393211460915303, 2.039758592881155, 2.426317196740967, 3.439752886203170, 4.106570214901189, 4.804701969809159, 5.294226680249339, 5.957404054878986, 6.482960837011903, 7.128719533342675, 7.829288998111871, 8.173443943169182, 8.878739252166391, 9.756087280135277, 10.04618649388703, 10.50532650248705, 11.26690806255471, 11.62611689580867, 12.26832834445547, 12.71773018306505, 13.25325350561011, 13.77682373761716, 14.39231318351811, 14.98960377008557, 15.43626471374847

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