Properties

Label 2-2940-1.1-c1-0-4
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 − 4·13-s − 15-s − 2·17-s − 2·19-s + 4·23-s + 25-s + 27-s + 6·29-s + 2·31-s − 2·33-s + 10·37-s − 4·39-s + 10·41-s + 12·43-s − 45-s + 8·47-s − 2·51-s + 2·55-s − 2·57-s + 8·59-s + 2·61-s + 4·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s + 1.64·37-s − 0.640·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 0.269·55-s − 0.264·57-s + 1.04·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.850313331\)
\(L(\frac12)\) \(\approx\) \(1.850313331\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−8.836199545630867645438721125526, −7.82067649776875631788978756567, −7.51325888882796800062561076037, −6.62104364944396067563104383367, −5.67017038367597175775538292471, −4.60737339403970738856758018331, −4.18636216199252309981165234200, −2.80785568081983867387706785012, −2.44981952348987740648309403533, −0.798786744184690487080656808366, 0.798786744184690487080656808366, 2.44981952348987740648309403533, 2.80785568081983867387706785012, 4.18636216199252309981165234200, 4.60737339403970738856758018331, 5.67017038367597175775538292471, 6.62104364944396067563104383367, 7.51325888882796800062561076037, 7.82067649776875631788978756567, 8.836199545630867645438721125526

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