Properties

Label 2-3040-1.1-c1-0-52
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  + 3.27·3-s + 5-s + 3.12·7-s + 7.71·9-s − 2.17·11-s + 0.388·13-s + 3.27·15-s + 0.537·17-s + 19-s + 10.2·21-s + 3.12·23-s + 25-s + 15.4·27-s + 2.13·29-s − 10.3·31-s − 7.13·33-s + 3.12·35-s − 6.91·37-s + 1.27·39-s − 5.43·41-s + 3.25·43-s + 7.71·45-s − 9.88·47-s + 2.76·49-s + 1.76·51-s − 2.68·53-s − 2.17·55-s + ⋯
L(s)  = 1  + 1.89·3-s + 0.447·5-s + 1.18·7-s + 2.57·9-s − 0.657·11-s + 0.107·13-s + 0.845·15-s + 0.130·17-s + 0.229·19-s + 2.23·21-s + 0.651·23-s + 0.200·25-s + 2.97·27-s + 0.395·29-s − 1.86·31-s − 1.24·33-s + 0.528·35-s − 1.13·37-s + 0.203·39-s − 0.848·41-s + 0.496·43-s + 1.15·45-s − 1.44·47-s + 0.394·49-s + 0.246·51-s − 0.369·53-s − 0.293·55-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

\(L(1)\) \(\approx\) \(4.719723869\)
\(L(\frac12)\) \(\approx\) \(4.719723869\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.27T + 3T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + 2.17T + 11T^{2} \)
13 \( 1 - 0.388T + 13T^{2} \)
17 \( 1 - 0.537T + 17T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 - 2.13T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 + 5.43T + 41T^{2} \)
43 \( 1 - 3.25T + 43T^{2} \)
47 \( 1 + 9.88T + 47T^{2} \)
53 \( 1 + 2.68T + 53T^{2} \)
59 \( 1 - 9.67T + 59T^{2} \)
61 \( 1 + 9.50T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 7.43T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + 2.25T + 97T^{2} \)
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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

−8.506117420904140992057850770773, −8.228523212823609632482465084492, −7.40479634790083874244279086420, −6.84843748186837083408819276277, −5.40057122921535109700431073562, −4.79379913781776085624608256768, −3.76845586096931487189620764213, −3.01320296815060661689465372901, −2.07986936407284526120542212678, −1.46329630058578742818697629198, 1.46329630058578742818697629198, 2.07986936407284526120542212678, 3.01320296815060661689465372901, 3.76845586096931487189620764213, 4.79379913781776085624608256768, 5.40057122921535109700431073562, 6.84843748186837083408819276277, 7.40479634790083874244279086420, 8.228523212823609632482465084492, 8.506117420904140992057850770773

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