Properties

Label 2-4140-1.1-c1-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3.44·7-s + 2.44·11-s + 4.44·13-s − 5.44·17-s − 1.55·19-s − 23-s + 25-s − 1.89·29-s − 7·31-s − 3.44·35-s + 6.34·37-s − 7.89·41-s − 8.89·43-s + 2.44·47-s + 4.89·49-s + 4.34·53-s + 2.44·55-s + 1.89·59-s − 5.34·61-s + 4.44·65-s + 1.44·67-s − 3·71-s + 9.34·73-s − 8.44·77-s − 4·79-s + 10.3·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.30·7-s + 0.738·11-s + 1.23·13-s − 1.32·17-s − 0.355·19-s − 0.208·23-s + 0.200·25-s − 0.352·29-s − 1.25·31-s − 0.583·35-s + 1.04·37-s − 1.23·41-s − 1.35·43-s + 0.357·47-s + 0.699·49-s + 0.597·53-s + 0.330·55-s + 0.247·59-s − 0.684·61-s + 0.551·65-s + 0.177·67-s − 0.356·71-s + 1.09·73-s − 0.962·77-s − 0.450·79-s + 1.13·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 + 5.44T + 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
29 \( 1 + 1.89T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 6.34T + 37T^{2} \)
41 \( 1 + 7.89T + 41T^{2} \)
43 \( 1 + 8.89T + 43T^{2} \)
47 \( 1 - 2.44T + 47T^{2} \)
53 \( 1 - 4.34T + 53T^{2} \)
59 \( 1 - 1.89T + 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 - 1.44T + 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.215062155958794478784712367833, −7.05422398999643817662684335237, −6.46675581517503817474233888395, −6.12401257948697757839470333869, −5.14539550830265368180116669930, −3.99693223423124269570719343192, −3.54387774498123272103179296335, −2.45978184089193883410605721198, −1.45222310834880214385841446354, 0, 1.45222310834880214385841446354, 2.45978184089193883410605721198, 3.54387774498123272103179296335, 3.99693223423124269570719343192, 5.14539550830265368180116669930, 6.12401257948697757839470333869, 6.46675581517503817474233888395, 7.05422398999643817662684335237, 8.215062155958794478784712367833

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