Properties

Label 2-3720-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 1.09·7-s + 9-s + 5.25·11-s + 1.24·13-s + 15-s − 2.64·17-s + 3.25·19-s + 1.09·21-s − 6.99·23-s + 25-s − 27-s + 5.89·29-s + 31-s − 5.25·33-s + 1.09·35-s − 3.24·37-s − 1.24·39-s + 4·41-s − 4.34·43-s − 45-s + 7.14·47-s − 5.80·49-s + 2.64·51-s + 6.80·53-s − 5.25·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.413·7-s + 0.333·9-s + 1.58·11-s + 0.346·13-s + 0.258·15-s − 0.641·17-s + 0.747·19-s + 0.238·21-s − 1.45·23-s + 0.200·25-s − 0.192·27-s + 1.09·29-s + 0.179·31-s − 0.915·33-s + 0.185·35-s − 0.534·37-s − 0.200·39-s + 0.624·41-s − 0.662·43-s − 0.149·45-s + 1.04·47-s − 0.828·49-s + 0.370·51-s + 0.934·53-s − 0.708·55-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: \(0\)
Selberg data: \((2,\ 3720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.359398291\)
\(L(\frac12)\) \(\approx\) \(1.359398291\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.09T + 7T^{2} \)
11 \( 1 - 5.25T + 11T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 - 3.25T + 19T^{2} \)
23 \( 1 + 6.99T + 23T^{2} \)
29 \( 1 - 5.89T + 29T^{2} \)
37 \( 1 + 3.24T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 4.34T + 43T^{2} \)
47 \( 1 - 7.14T + 47T^{2} \)
53 \( 1 - 6.80T + 53T^{2} \)
59 \( 1 + 8.80T + 59T^{2} \)
61 \( 1 - 2.91T + 61T^{2} \)
67 \( 1 - 1.43T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 8.69T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 0.0433T + 83T^{2} \)
89 \( 1 - 4.63T + 89T^{2} \)
97 \( 1 - 0.309T + 97T^{2} \)
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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

−8.582700506809040013183590541459, −7.72428913503810348554491418798, −6.84507741587962547329411037434, −6.38449272942826358006487972263, −5.67926337164662786597921635712, −4.55895239649700015890287367839, −4.00025442006870157291092781782, −3.17552935359166450437443391236, −1.81506559997777465915650910921, −0.71666207466301013753571570161, 0.71666207466301013753571570161, 1.81506559997777465915650910921, 3.17552935359166450437443391236, 4.00025442006870157291092781782, 4.55895239649700015890287367839, 5.67926337164662786597921635712, 6.38449272942826358006487972263, 6.84507741587962547329411037434, 7.72428913503810348554491418798, 8.582700506809040013183590541459

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