Properties

Label 2-4140-1.1-c1-0-3
Degree $2$
Conductor $4140$
Sign $1$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + 5-s − 3.73·7-s − 4.84·11-s − 2.84·13-s − 0.890·17-s + 6.84·19-s − 23-s + 25-s + 0.890·29-s + 7.73·31-s − 3.73·35-s − 1.95·37-s − 12.3·41-s − 3.47·43-s + 6.62·47-s + 6.95·49-s − 12.3·53-s − 4.84·55-s − 0.890·59-s + 8.62·61-s − 2.84·65-s + 7.73·67-s + 12.3·71-s + 16.5·73-s + 18.0·77-s + 13.4·79-s + 4.58·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.41·7-s − 1.46·11-s − 0.788·13-s − 0.216·17-s + 1.57·19-s − 0.208·23-s + 0.200·25-s + 0.165·29-s + 1.38·31-s − 0.631·35-s − 0.321·37-s − 1.93·41-s − 0.529·43-s + 0.966·47-s + 0.993·49-s − 1.69·53-s − 0.653·55-s − 0.115·59-s + 1.10·61-s − 0.352·65-s + 0.945·67-s + 1.46·71-s + 1.93·73-s + 2.06·77-s + 1.51·79-s + 0.502·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.195634843\)
\(L(\frac12)\) \(\approx\) \(1.195634843\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 + 3.73T + 7T^{2} \)
11 \( 1 + 4.84T + 11T^{2} \)
13 \( 1 + 2.84T + 13T^{2} \)
17 \( 1 + 0.890T + 17T^{2} \)
19 \( 1 - 6.84T + 19T^{2} \)
29 \( 1 - 0.890T + 29T^{2} \)
31 \( 1 - 7.73T + 31T^{2} \)
37 \( 1 + 1.95T + 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 + 3.47T + 43T^{2} \)
47 \( 1 - 6.62T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 0.890T + 59T^{2} \)
61 \( 1 - 8.62T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 4.58T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 17.2T + 97T^{2} \)
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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

−8.315980577922586045499568997226, −7.71202951961405956826639303019, −6.82168432893485571516960758665, −6.35782044232975906956561047434, −5.25367606936592739201464919667, −5.01649557363182209801530466337, −3.56846794282109298201789725206, −2.94138419891333984158036316766, −2.18772630102367677477184677524, −0.58914337766637151662930508360, 0.58914337766637151662930508360, 2.18772630102367677477184677524, 2.94138419891333984158036316766, 3.56846794282109298201789725206, 5.01649557363182209801530466337, 5.25367606936592739201464919667, 6.35782044232975906956561047434, 6.82168432893485571516960758665, 7.71202951961405956826639303019, 8.315980577922586045499568997226

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