Properties

Label 2-2940-1.1-c1-0-15
Degree $2$
Conductor $2940$
Sign $-1$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 1.64·11-s + 2.64·13-s + 15-s + 1.64·17-s − 8.29·19-s + 1.64·23-s + 25-s − 27-s + 7.64·29-s + 4.29·31-s + 1.64·33-s + 0.645·37-s − 2.64·39-s − 4.93·41-s − 5.93·43-s − 45-s − 6·47-s − 1.64·51-s + 3.29·53-s + 1.64·55-s + 8.29·57-s − 10.9·59-s − 8·61-s − 2.64·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.333·9-s − 0.496·11-s + 0.733·13-s + 0.258·15-s + 0.399·17-s − 1.90·19-s + 0.343·23-s + 0.200·25-s − 0.192·27-s + 1.41·29-s + 0.770·31-s + 0.286·33-s + 0.106·37-s − 0.423·39-s − 0.771·41-s − 0.905·43-s − 0.149·45-s − 0.875·47-s − 0.230·51-s + 0.452·53-s + 0.221·55-s + 1.09·57-s − 1.42·59-s − 1.02·61-s − 0.328·65-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: no
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 \( 1 + 1.64T + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 + 8.29T + 19T^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
29 \( 1 - 7.64T + 29T^{2} \)
31 \( 1 - 4.29T + 31T^{2} \)
37 \( 1 - 0.645T + 37T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + 5.93T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 0.645T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 2.29T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−8.307462494790344301334692836849, −7.75112764928530717455200414400, −6.47866242269807619472835415173, −6.43036861165827495316228777643, −5.17164068998328957742613280428, −4.55429118265446667781241671394, −3.66199299500958599632960753517, −2.63450416022290959496984609206, −1.34341653690519261556267413256, 0, 1.34341653690519261556267413256, 2.63450416022290959496984609206, 3.66199299500958599632960753517, 4.55429118265446667781241671394, 5.17164068998328957742613280428, 6.43036861165827495316228777643, 6.47866242269807619472835415173, 7.75112764928530717455200414400, 8.307462494790344301334692836849

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