Properties

Label 2-29400-1.1-c1-0-110
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 − 2·11-s + 13-s + 8·17-s − 7·19-s + 6·23-s + 27-s + 4·29-s − 8·31-s − 2·33-s − 7·37-s + 39-s − 2·41-s − 4·43-s + 12·47-s + 8·51-s + 4·53-s − 7·57-s − 12·59-s − 3·61-s − 9·67-s + 6·69-s − 12·71-s + 9·73-s − 17·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 1.94·17-s − 1.60·19-s + 1.25·23-s + 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.348·33-s − 1.15·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 1.75·47-s + 1.12·51-s + 0.549·53-s − 0.927·57-s − 1.56·59-s − 0.384·61-s − 1.09·67-s + 0.722·69-s − 1.42·71-s + 1.05·73-s − 1.91·79-s + 1/9·81-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−15.23213049650338, −14.94475658538707, −14.42941554925489, −13.83513498237069, −13.37354292980906, −12.70947171516776, −12.39750660431971, −11.84058736853867, −10.93251792318186, −10.48010179053792, −10.24615943952999, −9.304620086889307, −8.933381275063855, −8.322208712240243, −7.830529827127246, −7.184467959886565, −6.742709856778752, −5.760536540040213, −5.459241122862947, −4.613139146780197, −3.984371601415935, −3.200437952712146, −2.822642873496331, −1.844847235525059, −1.189013528330304, 0, 1.189013528330304, 1.844847235525059, 2.822642873496331, 3.200437952712146, 3.984371601415935, 4.613139146780197, 5.459241122862947, 5.760536540040213, 6.742709856778752, 7.184467959886565, 7.830529827127246, 8.322208712240243, 8.933381275063855, 9.304620086889307, 10.24615943952999, 10.48010179053792, 10.93251792318186, 11.84058736853867, 12.39750660431971, 12.70947171516776, 13.37354292980906, 13.83513498237069, 14.42941554925489, 14.94475658538707, 15.23213049650338

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