Properties

Label 2-4140-1.1-c1-0-24
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 − 1.56·7-s − 2·11-s + 0.561·13-s + 1.56·17-s + 6·19-s − 23-s + 25-s + 2.12·29-s − 9.24·31-s + 1.56·35-s − 0.438·37-s + 4.12·41-s + 7.68·47-s − 4.56·49-s + 0.438·53-s + 2·55-s − 8.68·59-s + 1.12·61-s − 0.561·65-s − 4.43·67-s − 1.87·71-s − 8.56·73-s + 3.12·77-s + 13.1·79-s − 14.9·83-s − 1.56·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.590·7-s − 0.603·11-s + 0.155·13-s + 0.378·17-s + 1.37·19-s − 0.208·23-s + 0.200·25-s + 0.394·29-s − 1.66·31-s + 0.263·35-s − 0.0720·37-s + 0.643·41-s + 1.12·47-s − 0.651·49-s + 0.0602·53-s + 0.269·55-s − 1.13·59-s + 0.143·61-s − 0.0696·65-s − 0.542·67-s − 0.222·71-s − 1.00·73-s + 0.355·77-s + 1.47·79-s − 1.63·83-s − 0.169·85-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 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 + 9.24T + 31T^{2} \)
37 \( 1 + 0.438T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 - 0.438T + 53T^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 - 1.12T + 61T^{2} \)
67 \( 1 + 4.43T + 67T^{2} \)
71 \( 1 + 1.87T + 71T^{2} \)
73 \( 1 + 8.56T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 - 2.24T + 89T^{2} \)
97 \( 1 + 4.87T + 97T^{2} \)
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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

−7.82142074295995730408799812098, −7.50417492171943738086114574144, −6.65115116690813656959056279803, −5.73940719674188407132664032847, −5.18359000576561728478540914690, −4.13923595809106487854865692892, −3.36480036572835287791980102643, −2.65496218881759201842139255484, −1.31189355048478474088904992157, 0, 1.31189355048478474088904992157, 2.65496218881759201842139255484, 3.36480036572835287791980102643, 4.13923595809106487854865692892, 5.18359000576561728478540914690, 5.73940719674188407132664032847, 6.65115116690813656959056279803, 7.50417492171943738086114574144, 7.82142074295995730408799812098

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