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 − 2.61·3-s + 4-s − 1.38·5-s − 2.61·6-s − 3·7-s + 8-s + 3.85·9-s − 1.38·10-s − 2.38·11-s − 2.61·12-s − 3·13-s − 3·14-s + 3.61·15-s + 16-s − 7.47·17-s + 3.85·18-s − 3·19-s − 1.38·20-s + 7.85·21-s − 2.38·22-s + 1.47·23-s − 2.61·24-s − 3.09·25-s − 3·26-s − 2.23·27-s − 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.51·3-s + 0.5·4-s − 0.618·5-s − 1.06·6-s − 1.13·7-s + 0.353·8-s + 1.28·9-s − 0.437·10-s − 0.718·11-s − 0.755·12-s − 0.832·13-s − 0.801·14-s + 0.934·15-s + 0.250·16-s − 1.81·17-s + 0.908·18-s − 0.688·19-s − 0.309·20-s + 1.71·21-s − 0.507·22-s + 0.306·23-s − 0.534·24-s − 0.618·25-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 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 + 1.38T + 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 2.38T + 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + 7.47T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 - 1.47T + 23T^{2} \)
29 \( 1 + 9.23T + 29T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 1.85T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 9.23T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 9.76T + 59T^{2} \)
61 \( 1 + 2.38T + 61T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 0.708T + 73T^{2} \)
79 \( 1 + 1.85T + 79T^{2} \)
83 \( 1 + 1.94T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−7.20193873697552341772487463710, −6.45082350381719454975173900019, −5.93825411806962806963479187343, −5.25737649908674734961429262711, −4.49115166715489506444535199749, −3.94922063529863278727306577464, −2.89493040810100742624055535855, −1.94434150521789333176918400896, 0, 0, 1.94434150521789333176918400896, 2.89493040810100742624055535855, 3.94922063529863278727306577464, 4.49115166715489506444535199749, 5.25737649908674734961429262711, 5.93825411806962806963479187343, 6.45082350381719454975173900019, 7.20193873697552341772487463710

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