Properties

Label 2-4140-1.1-c1-0-2
Degree $2$
Conductor $4140$
Sign $1$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 2·13-s − 6·17-s + 2·19-s + 23-s + 25-s − 6·29-s − 4·31-s + 4·35-s + 8·37-s − 6·41-s + 8·43-s − 12·47-s + 9·49-s + 6·53-s + 6·59-s − 10·61-s − 2·65-s + 8·67-s + 6·71-s + 2·73-s − 10·79-s + 12·83-s + 6·85-s − 6·89-s − 8·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.676·35-s + 1.31·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.781·59-s − 1.28·61-s − 0.248·65-s + 0.977·67-s + 0.712·71-s + 0.234·73-s − 1.12·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.838·91-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9747785660\)
\(L(\frac12)\) \(\approx\) \(0.9747785660\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.511348744562729280220841628695, −7.59364285167981924845250143718, −6.88272774995945306399970033031, −6.32933415016630436674097688106, −5.58414743690763122838651332783, −4.52179776183978086384556961510, −3.71699009524032920517802328202, −3.11009796581965296703483268985, −2.05836789002722838619797179361, −0.53754359514065542070459562379, 0.53754359514065542070459562379, 2.05836789002722838619797179361, 3.11009796581965296703483268985, 3.71699009524032920517802328202, 4.52179776183978086384556961510, 5.58414743690763122838651332783, 6.32933415016630436674097688106, 6.88272774995945306399970033031, 7.59364285167981924845250143718, 8.511348744562729280220841628695

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