Properties

Label 2-3720-1.1-c1-0-47
Degree $2$
Conductor $3720$
Sign $-1$
Analytic cond. $29.7043$
Root an. cond. $5.45016$
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 − 2.48·7-s + 9-s − 0.387·11-s + 2.48·13-s − 15-s − 0.418·17-s − 4·19-s − 2.48·21-s + 5.11·23-s + 25-s + 27-s − 4.24·29-s − 31-s − 0.387·33-s + 2.48·35-s + 3.83·37-s + 2.48·39-s + 4.31·41-s − 6·43-s − 45-s − 10.8·47-s − 0.843·49-s − 0.418·51-s − 8.85·53-s + 0.387·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.937·7-s + 0.333·9-s − 0.116·11-s + 0.688·13-s − 0.258·15-s − 0.101·17-s − 0.917·19-s − 0.541·21-s + 1.06·23-s + 0.200·25-s + 0.192·27-s − 0.789·29-s − 0.179·31-s − 0.0675·33-s + 0.419·35-s + 0.629·37-s + 0.397·39-s + 0.673·41-s − 0.914·43-s − 0.149·45-s − 1.58·47-s − 0.120·49-s − 0.0585·51-s − 1.21·53-s + 0.0523·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3720\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-1$
Analytic conductor: \(29.7043\)
Root analytic conductor: \(5.45016\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3720,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
31 \( 1 + T \)
good7 \( 1 + 2.48T + 7T^{2} \)
11 \( 1 + 0.387T + 11T^{2} \)
13 \( 1 - 2.48T + 13T^{2} \)
17 \( 1 + 0.418T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
37 \( 1 - 3.83T + 37T^{2} \)
41 \( 1 - 4.31T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 + 3.86T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 - 7.59T + 71T^{2} \)
73 \( 1 + 4.60T + 73T^{2} \)
79 \( 1 + 5.38T + 79T^{2} \)
83 \( 1 + 5.89T + 83T^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−8.237346912691095282643228644248, −7.45438476595171983408116883124, −6.67860909478040035067302040486, −6.12048277553750258296160034511, −5.02457931357527338414866918336, −4.12450733998618051804368024320, −3.39947419362783258276354749745, −2.71639421188040218634791006690, −1.48443147360890699640388357368, 0, 1.48443147360890699640388357368, 2.71639421188040218634791006690, 3.39947419362783258276354749745, 4.12450733998618051804368024320, 5.02457931357527338414866918336, 6.12048277553750258296160034511, 6.67860909478040035067302040486, 7.45438476595171983408116883124, 8.237346912691095282643228644248

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