Properties

Label 2-2940-1.1-c1-0-26
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 4·11-s − 7·13-s + 15-s + 6·17-s − 3·19-s − 2·23-s + 25-s + 27-s − 2·29-s − 7·31-s − 4·33-s − 7·37-s − 7·39-s + 8·41-s + 5·43-s + 45-s − 10·47-s + 6·51-s − 8·53-s − 4·55-s − 3·57-s − 10·59-s + 6·61-s − 7·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.94·13-s + 0.258·15-s + 1.45·17-s − 0.688·19-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.25·31-s − 0.696·33-s − 1.15·37-s − 1.12·39-s + 1.24·41-s + 0.762·43-s + 0.149·45-s − 1.45·47-s + 0.840·51-s − 1.09·53-s − 0.539·55-s − 0.397·57-s − 1.30·59-s + 0.768·61-s − 0.868·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−8.264234219052835662840524102271, −7.58038405978316471743308736553, −7.19601002521936360224755429218, −5.94982713858785099491107604520, −5.25972398836122273924768521712, −4.56272009636257024202589978944, −3.35264561435067538664390565583, −2.57901384661548545444883920131, −1.78693567527845542392800100347, 0, 1.78693567527845542392800100347, 2.57901384661548545444883920131, 3.35264561435067538664390565583, 4.56272009636257024202589978944, 5.25972398836122273924768521712, 5.94982713858785099491107604520, 7.19601002521936360224755429218, 7.58038405978316471743308736553, 8.264234219052835662840524102271

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