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Properties

Label 2-29400-1.1-c1-0-122
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 6·11-s − 7·13-s − 4·17-s − 5·19-s − 6·23-s + 27-s − 4·29-s − 8·31-s − 6·33-s − 37-s − 7·39-s − 2·41-s − 4·43-s + 8·47-s − 4·51-s − 8·53-s − 5·57-s − 5·61-s + 5·67-s − 6·69-s + 4·71-s + 73-s + 11·79-s + 81-s − 16·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.94·13-s − 0.970·17-s − 1.14·19-s − 1.25·23-s + 0.192·27-s − 0.742·29-s − 1.43·31-s − 1.04·33-s − 0.164·37-s − 1.12·39-s − 0.312·41-s − 0.609·43-s + 1.16·47-s − 0.560·51-s − 1.09·53-s − 0.662·57-s − 0.640·61-s + 0.610·67-s − 0.722·69-s + 0.474·71-s + 0.117·73-s + 1.23·79-s + 1/9·81-s − 1.75·83-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: \(2\)
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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−15.57033278491615, −15.07807461570330, −14.79766250294245, −14.07964818021801, −13.59807312311374, −12.97797761353631, −12.56818859589404, −12.25004272608200, −11.28136548206618, −10.80125635211278, −10.24851175058241, −9.810694831337052, −9.209728184034638, −8.583458586401721, −7.933021908804984, −7.576872322294947, −7.050064661314379, −6.308593975536494, −5.471313839900285, −5.007020715293029, −4.396776708926505, −3.704134310978860, −2.780646127182824, −2.244300445420809, −1.916043584411209, 0, 0, 1.916043584411209, 2.244300445420809, 2.780646127182824, 3.704134310978860, 4.396776708926505, 5.007020715293029, 5.471313839900285, 6.308593975536494, 7.050064661314379, 7.576872322294947, 7.933021908804984, 8.583458586401721, 9.209728184034638, 9.810694831337052, 10.24851175058241, 10.80125635211278, 11.28136548206618, 12.25004272608200, 12.56818859589404, 12.97797761353631, 13.59807312311374, 14.07964818021801, 14.79766250294245, 15.07807461570330, 15.57033278491615

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