Properties

Label 2-3040-1.1-c1-0-15
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  + 1.09·3-s − 5-s − 2.80·7-s − 1.80·9-s + 2.45·11-s + 3.55·13-s − 1.09·15-s − 4.34·17-s − 19-s − 3.06·21-s + 1.19·23-s + 25-s − 5.25·27-s + 7.44·29-s + 2.91·31-s + 2.68·33-s + 2.80·35-s + 7.89·37-s + 3.88·39-s − 10.5·41-s + 2.95·43-s + 1.80·45-s + 13.1·47-s + 0.845·49-s − 4.75·51-s + 3.74·53-s − 2.45·55-s + ⋯
L(s)  = 1  + 0.632·3-s − 0.447·5-s − 1.05·7-s − 0.600·9-s + 0.740·11-s + 0.984·13-s − 0.282·15-s − 1.05·17-s − 0.229·19-s − 0.669·21-s + 0.250·23-s + 0.200·25-s − 1.01·27-s + 1.38·29-s + 0.523·31-s + 0.468·33-s + 0.473·35-s + 1.29·37-s + 0.622·39-s − 1.64·41-s + 0.450·43-s + 0.268·45-s + 1.91·47-s + 0.120·49-s − 0.666·51-s + 0.513·53-s − 0.331·55-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

\(L(1)\) \(\approx\) \(1.719270089\)
\(L(\frac12)\) \(\approx\) \(1.719270089\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.09T + 3T^{2} \)
7 \( 1 + 2.80T + 7T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 - 3.55T + 13T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
23 \( 1 - 1.19T + 23T^{2} \)
29 \( 1 - 7.44T + 29T^{2} \)
31 \( 1 - 2.91T + 31T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 2.45T + 61T^{2} \)
67 \( 1 + 9.59T + 67T^{2} \)
71 \( 1 + 7.60T + 71T^{2} \)
73 \( 1 - 2.75T + 73T^{2} \)
79 \( 1 - 9.79T + 79T^{2} \)
83 \( 1 - 6.95T + 83T^{2} \)
89 \( 1 - 8.49T + 89T^{2} \)
97 \( 1 - 2.01T + 97T^{2} \)
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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

−8.778857774679386445825970822939, −8.197415859258520608683184467275, −7.16501924952996723643791733072, −6.43008187213180767383469118791, −5.94969110076985092628436029925, −4.62326830523207137709662596325, −3.81818948258221125358358272104, −3.15245234625981408337246521820, −2.29912738106816343959111198444, −0.76040197716327222908052154906, 0.76040197716327222908052154906, 2.29912738106816343959111198444, 3.15245234625981408337246521820, 3.81818948258221125358358272104, 4.62326830523207137709662596325, 5.94969110076985092628436029925, 6.43008187213180767383469118791, 7.16501924952996723643791733072, 8.197415859258520608683184467275, 8.778857774679386445825970822939

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