Properties

Label 2-3040-1.1-c1-0-13
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  + 5-s − 2.82·7-s − 3·9-s + 4·11-s − 0.828·13-s − 3.65·17-s − 19-s − 2.82·23-s + 25-s + 7.65·29-s + 5.65·31-s − 2.82·35-s + 10.4·37-s + 7.65·41-s − 8.48·43-s − 3·45-s − 10.8·47-s + 1.00·49-s + 12.8·53-s + 4·55-s + 1.65·59-s + 6·61-s + 8.48·63-s − 0.828·65-s + 11.3·67-s + 5.65·71-s + 4.34·73-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.06·7-s − 9-s + 1.20·11-s − 0.229·13-s − 0.886·17-s − 0.229·19-s − 0.589·23-s + 0.200·25-s + 1.42·29-s + 1.01·31-s − 0.478·35-s + 1.72·37-s + 1.19·41-s − 1.29·43-s − 0.447·45-s − 1.57·47-s + 0.142·49-s + 1.76·53-s + 0.539·55-s + 0.215·59-s + 0.768·61-s + 1.06·63-s − 0.102·65-s + 1.38·67-s + 0.671·71-s + 0.508·73-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

\(L(1)\) \(\approx\) \(1.534401944\)
\(L(\frac12)\) \(\approx\) \(1.534401944\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 4.34T + 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 0.343T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−8.660588447112110479282914369416, −8.241020644193179124639666027941, −6.90682651057626838526529177761, −6.40350774393606988538874210083, −5.95623869162944919487791266276, −4.81795867000703910332542638572, −3.95875585348658232981967543128, −2.98917693948920840550455376871, −2.24313892026549684212625283889, −0.73582021518784170435792738917, 0.73582021518784170435792738917, 2.24313892026549684212625283889, 2.98917693948920840550455376871, 3.95875585348658232981967543128, 4.81795867000703910332542638572, 5.95623869162944919487791266276, 6.40350774393606988538874210083, 6.90682651057626838526529177761, 8.241020644193179124639666027941, 8.660588447112110479282914369416

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