Properties

Label 2-29400-1.1-c1-0-88
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 − 5·11-s − 2·13-s + 6·17-s − 2·19-s + 5·23-s + 27-s − 5·29-s − 4·31-s − 5·33-s + 37-s − 2·39-s − 12·41-s + 5·43-s + 2·47-s + 6·51-s + 14·53-s − 2·57-s + 2·59-s − 5·67-s + 5·69-s − 9·71-s + 10·73-s + 11·79-s + 81-s + 16·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.04·23-s + 0.192·27-s − 0.928·29-s − 0.718·31-s − 0.870·33-s + 0.164·37-s − 0.320·39-s − 1.87·41-s + 0.762·43-s + 0.291·47-s + 0.840·51-s + 1.92·53-s − 0.264·57-s + 0.260·59-s − 0.610·67-s + 0.601·69-s − 1.06·71-s + 1.17·73-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: $\chi_{29400} (1, \cdot )$
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.17793731781928, −14.95443462141668, −14.55443978261040, −13.68220573176656, −13.37410157589181, −12.86965226334995, −12.27200406951012, −11.88124294580780, −10.86533221384382, −10.67053020642405, −9.961277047140201, −9.577435689293223, −8.817013287820100, −8.353809916895013, −7.587248580158297, −7.452091702026214, −6.721852835991608, −5.755570621944582, −5.288545346915580, −4.841960854945648, −3.853701937142902, −3.324077153722310, −2.613666625159299, −2.070606478414251, −1.062619094441708, 0, 1.062619094441708, 2.070606478414251, 2.613666625159299, 3.324077153722310, 3.853701937142902, 4.841960854945648, 5.288545346915580, 5.755570621944582, 6.721852835991608, 7.452091702026214, 7.587248580158297, 8.353809916895013, 8.817013287820100, 9.577435689293223, 9.961277047140201, 10.67053020642405, 10.86533221384382, 11.88124294580780, 12.27200406951012, 12.86965226334995, 13.37410157589181, 13.68220573176656, 14.55443978261040, 14.95443462141668, 15.17793731781928

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