Properties

Label 2-3040-1.1-c1-0-66
Degree $2$
Conductor $3040$
Sign $-1$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
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  + 2.19·3-s − 5-s − 0.827·7-s + 1.82·9-s + 1.74·11-s − 3.94·13-s − 2.19·15-s − 4.92·17-s + 19-s − 1.81·21-s − 4.82·23-s + 25-s − 2.57·27-s − 6.97·29-s + 5.49·31-s + 3.84·33-s + 0.827·35-s + 0.975·37-s − 8.66·39-s + 5.15·41-s − 6.48·43-s − 1.82·45-s − 10.0·47-s − 6.31·49-s − 10.8·51-s − 10.3·53-s − 1.74·55-s + ⋯
L(s)  = 1  + 1.26·3-s − 0.447·5-s − 0.312·7-s + 0.609·9-s + 0.527·11-s − 1.09·13-s − 0.567·15-s − 1.19·17-s + 0.229·19-s − 0.396·21-s − 1.00·23-s + 0.200·25-s − 0.495·27-s − 1.29·29-s + 0.987·31-s + 0.668·33-s + 0.139·35-s + 0.160·37-s − 1.38·39-s + 0.804·41-s − 0.989·43-s − 0.272·45-s − 1.47·47-s − 0.902·49-s − 1.51·51-s − 1.42·53-s − 0.235·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 - 2.19T + 3T^{2} \)
7 \( 1 + 0.827T + 7T^{2} \)
11 \( 1 - 1.74T + 11T^{2} \)
13 \( 1 + 3.94T + 13T^{2} \)
17 \( 1 + 4.92T + 17T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 6.97T + 29T^{2} \)
31 \( 1 - 5.49T + 31T^{2} \)
37 \( 1 - 0.975T + 37T^{2} \)
41 \( 1 - 5.15T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 1.74T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 - 0.344T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 4.04T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−8.231443055807578888314815206085, −7.83832091407086557846736086823, −6.95945207939923327267807412695, −6.28979584302188614158725544481, −5.06666762529730812252551471033, −4.20121716683316481391955119021, −3.49772456487021535195278252685, −2.62608282104647283928607884271, −1.83274378701399444687965125692, 0, 1.83274378701399444687965125692, 2.62608282104647283928607884271, 3.49772456487021535195278252685, 4.20121716683316481391955119021, 5.06666762529730812252551471033, 6.28979584302188614158725544481, 6.95945207939923327267807412695, 7.83832091407086557846736086823, 8.231443055807578888314815206085

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