Properties

Label 2-3040-1.1-c1-0-41
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.78·3-s − 5-s + 3.76·7-s + 4.76·9-s − 1.15·11-s + 1.63·13-s − 2.78·15-s − 1.38·17-s − 19-s + 10.4·21-s + 7.76·23-s + 25-s + 4.92·27-s + 0.651·29-s − 4.31·31-s − 3.22·33-s − 3.76·35-s + 3.02·37-s + 4.54·39-s + 9.84·41-s − 9.95·43-s − 4.76·45-s + 3.62·47-s + 7.18·49-s − 3.87·51-s + 5.20·53-s + 1.15·55-s + ⋯
L(s)  = 1  + 1.60·3-s − 0.447·5-s + 1.42·7-s + 1.58·9-s − 0.348·11-s + 0.452·13-s − 0.719·15-s − 0.337·17-s − 0.229·19-s + 2.29·21-s + 1.61·23-s + 0.200·25-s + 0.947·27-s + 0.121·29-s − 0.774·31-s − 0.560·33-s − 0.636·35-s + 0.496·37-s + 0.727·39-s + 1.53·41-s − 1.51·43-s − 0.710·45-s + 0.528·47-s + 1.02·49-s − 0.542·51-s + 0.714·53-s + 0.155·55-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: \(0\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.790971799\)
\(L(\frac12)\) \(\approx\) \(3.790971799\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.78T + 3T^{2} \)
7 \( 1 - 3.76T + 7T^{2} \)
11 \( 1 + 1.15T + 11T^{2} \)
13 \( 1 - 1.63T + 13T^{2} \)
17 \( 1 + 1.38T + 17T^{2} \)
23 \( 1 - 7.76T + 23T^{2} \)
29 \( 1 - 0.651T + 29T^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
37 \( 1 - 3.02T + 37T^{2} \)
41 \( 1 - 9.84T + 41T^{2} \)
43 \( 1 + 9.95T + 43T^{2} \)
47 \( 1 - 3.62T + 47T^{2} \)
53 \( 1 - 5.20T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 - 1.15T + 61T^{2} \)
67 \( 1 + 1.99T + 67T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 - 1.87T + 73T^{2} \)
79 \( 1 - 0.0412T + 79T^{2} \)
83 \( 1 + 5.95T + 83T^{2} \)
89 \( 1 + 0.794T + 89T^{2} \)
97 \( 1 + 18.9T + 97T^{2} \)
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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

−8.702786193712360056894408310926, −8.025432185926102537108665591332, −7.54579308125836003352306575755, −6.80584576440135399841016533073, −5.45864757697308698381448172102, −4.61781691988556818517367887222, −3.93416559270493762085704377476, −3.00834377558978269323652494966, −2.19805967985969834514527739630, −1.22434621387784133529060562152, 1.22434621387784133529060562152, 2.19805967985969834514527739630, 3.00834377558978269323652494966, 3.93416559270493762085704377476, 4.61781691988556818517367887222, 5.45864757697308698381448172102, 6.80584576440135399841016533073, 7.54579308125836003352306575755, 8.025432185926102537108665591332, 8.702786193712360056894408310926

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