Properties

Label 2-29400-1.1-c1-0-71
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 + 13-s − 6·17-s + 19-s + 2·23-s − 27-s − 3·37-s − 39-s − 8·43-s + 2·47-s + 6·51-s − 2·53-s − 57-s + 10·59-s + 5·61-s + 3·67-s − 2·69-s − 12·71-s + 13·73-s + 11·79-s + 81-s + 2·83-s + 6·89-s + 97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.277·13-s − 1.45·17-s + 0.229·19-s + 0.417·23-s − 0.192·27-s − 0.493·37-s − 0.160·39-s − 1.21·43-s + 0.291·47-s + 0.840·51-s − 0.274·53-s − 0.132·57-s + 1.30·59-s + 0.640·61-s + 0.366·67-s − 0.240·69-s − 1.42·71-s + 1.52·73-s + 1.23·79-s + 1/9·81-s + 0.219·83-s + 0.635·89-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-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: $\chi_{29400} (1, \cdot )$
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 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.62071829776572, −14.94578309937478, −14.42970684390931, −13.68774824284368, −13.25586917936455, −12.88032921765075, −12.13459985819259, −11.64525405428072, −11.19105090638548, −10.64330481258715, −10.17496585270955, −9.445249377785392, −8.944583490632639, −8.373943883363397, −7.735230854434167, −6.921371472167972, −6.651723008147709, −6.005285931374416, −5.208782894816667, −4.853177585822167, −4.042811205915730, −3.485457208804505, −2.536979473880521, −1.862726945460653, −0.9398448229058478, 0, 0.9398448229058478, 1.862726945460653, 2.536979473880521, 3.485457208804505, 4.042811205915730, 4.853177585822167, 5.208782894816667, 6.005285931374416, 6.651723008147709, 6.921371472167972, 7.735230854434167, 8.373943883363397, 8.944583490632639, 9.445249377785392, 10.17496585270955, 10.64330481258715, 11.19105090638548, 11.64525405428072, 12.13459985819259, 12.88032921765075, 13.25586917936455, 13.68774824284368, 14.42970684390931, 14.94578309937478, 15.62071829776572

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