Properties

Label 2-29400-1.1-c1-0-55
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·13-s + 6·17-s + 4·19-s + 4·23-s + 27-s + 6·29-s + 8·31-s + 10·37-s − 2·39-s + 10·41-s − 12·43-s − 8·47-s + 6·51-s − 6·53-s + 4·57-s − 4·59-s + 10·61-s − 12·67-s + 4·69-s + 4·71-s + 2·73-s + 8·79-s + 81-s + 4·83-s + 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.320·39-s + 1.56·41-s − 1.82·43-s − 1.16·47-s + 0.840·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s − 1.46·67-s + 0.481·69-s + 0.474·71-s + 0.234·73-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + 0.643·87-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: \(0\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.675576914\)
\(L(\frac12)\) \(\approx\) \(3.675576914\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.02347830736929, −14.60326821802695, −14.21238330784629, −13.59209760325311, −13.13319714884180, −12.48745511186442, −12.00695525292706, −11.52627279314355, −10.85358248506356, −10.06436680579987, −9.775331347208411, −9.353140773919024, −8.514842189242590, −7.947513658931576, −7.680518854097384, −6.890527222872790, −6.338087410124866, −5.607247597369898, −4.859260745567523, −4.506854807833460, −3.459987052697289, −3.026339168557931, −2.460344480659377, −1.356261749549409, −0.7921002605257131, 0.7921002605257131, 1.356261749549409, 2.460344480659377, 3.026339168557931, 3.459987052697289, 4.506854807833460, 4.859260745567523, 5.607247597369898, 6.338087410124866, 6.890527222872790, 7.680518854097384, 7.947513658931576, 8.514842189242590, 9.353140773919024, 9.775331347208411, 10.06436680579987, 10.85358248506356, 11.52627279314355, 12.00695525292706, 12.48745511186442, 13.13319714884180, 13.59209760325311, 14.21238330784629, 14.60326821802695, 15.02347830736929

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