Properties

Label 2-2940-1.1-c1-0-6
Degree $2$
Conductor $2940$
Sign $1$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 2·11-s − 13-s − 15-s + 4·17-s + 19-s + 4·23-s + 25-s + 27-s + 5·31-s − 2·33-s − 5·37-s − 39-s − 2·41-s − 9·43-s − 45-s + 2·47-s + 4·51-s + 12·53-s + 2·55-s + 57-s + 8·59-s + 14·61-s + 65-s + 9·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s + 0.970·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.898·31-s − 0.348·33-s − 0.821·37-s − 0.160·39-s − 0.312·41-s − 1.37·43-s − 0.149·45-s + 0.291·47-s + 0.560·51-s + 1.64·53-s + 0.269·55-s + 0.132·57-s + 1.04·59-s + 1.79·61-s + 0.124·65-s + 1.09·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2940} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.057059357\)
\(L(\frac12)\) \(\approx\) \(2.057059357\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.500459982889107791948072351983, −8.171950807751508916672808581419, −7.26411294117993636637552057855, −6.75853082244336358777403383052, −5.49024348463688223892917589201, −4.93300003544219171269645927792, −3.83714205863957343163612654118, −3.14294443805192686128138415785, −2.22601898104338938179843263310, −0.864305744297386540125109770640, 0.864305744297386540125109770640, 2.22601898104338938179843263310, 3.14294443805192686128138415785, 3.83714205863957343163612654118, 4.93300003544219171269645927792, 5.49024348463688223892917589201, 6.75853082244336358777403383052, 7.26411294117993636637552057855, 8.171950807751508916672808581419, 8.500459982889107791948072351983

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