Properties

Degree 2
Conductor $ 2 \cdot 3019 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.381·3-s + 4-s − 3.61·5-s − 0.381·6-s − 3·7-s + 8-s − 2.85·9-s − 3.61·10-s − 4.61·11-s − 0.381·12-s − 3·13-s − 3·14-s + 1.38·15-s + 16-s + 1.47·17-s − 2.85·18-s − 3·19-s − 3.61·20-s + 1.14·21-s − 4.61·22-s − 7.47·23-s − 0.381·24-s + 8.09·25-s − 3·26-s + 2.23·27-s − 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.220·3-s + 0.5·4-s − 1.61·5-s − 0.155·6-s − 1.13·7-s + 0.353·8-s − 0.951·9-s − 1.14·10-s − 1.39·11-s − 0.110·12-s − 0.832·13-s − 0.801·14-s + 0.356·15-s + 0.250·16-s + 0.357·17-s − 0.672·18-s − 0.688·19-s − 0.809·20-s + 0.250·21-s − 0.984·22-s − 1.55·23-s − 0.0779·24-s + 1.61·25-s − 0.588·26-s + 0.430·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6038\)    =    \(2 \cdot 3019\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6038} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 6038,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3019\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3019\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3019 \( 1+O(T) \)
good3 \( 1 + 0.381T + 3T^{2} \)
5 \( 1 + 3.61T + 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 4.61T + 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 + 4.76T + 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 - 4.85T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + 8.56T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 4.61T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 - 5.94T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 - 7.79T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.49637706986989643634034703940, −6.46178064959430146484122118225, −6.04002451607903648633524029033, −4.99050022039682986454604642705, −4.56388580421297763359868069522, −3.37692936114172083921266340471, −3.22422217533988060994973477349, −2.20540295289471703522520176571, 0, 0, 2.20540295289471703522520176571, 3.22422217533988060994973477349, 3.37692936114172083921266340471, 4.56388580421297763359868069522, 4.99050022039682986454604642705, 6.04002451607903648633524029033, 6.46178064959430146484122118225, 7.49637706986989643634034703940

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