Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.76·5-s + 1.84·7-s + 9-s + 1.01·11-s + 4.04·13-s − 3.76·15-s − 1.59·17-s − 1.49·19-s − 1.84·21-s + 8.43·23-s + 9.19·25-s − 27-s + 0.429·29-s + 3.08·31-s − 1.01·33-s + 6.95·35-s + 4.40·37-s − 4.04·39-s − 5.20·41-s − 8.83·43-s + 3.76·45-s − 4.52·47-s − 3.59·49-s + 1.59·51-s + 11.5·53-s + 3.83·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.68·5-s + 0.697·7-s + 0.333·9-s + 0.306·11-s + 1.12·13-s − 0.972·15-s − 0.386·17-s − 0.343·19-s − 0.402·21-s + 1.75·23-s + 1.83·25-s − 0.192·27-s + 0.0797·29-s + 0.554·31-s − 0.177·33-s + 1.17·35-s + 0.723·37-s − 0.647·39-s − 0.813·41-s − 1.34·43-s + 0.561·45-s − 0.659·47-s − 0.513·49-s + 0.223·51-s + 1.58·53-s + 0.516·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.995136700$
$L(\frac12)$  $\approx$  $2.995136700$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
503 \( 1 + T \)
good5 \( 1 - 3.76T + 5T^{2} \)
7 \( 1 - 1.84T + 7T^{2} \)
11 \( 1 - 1.01T + 11T^{2} \)
13 \( 1 - 4.04T + 13T^{2} \)
17 \( 1 + 1.59T + 17T^{2} \)
19 \( 1 + 1.49T + 19T^{2} \)
23 \( 1 - 8.43T + 23T^{2} \)
29 \( 1 - 0.429T + 29T^{2} \)
31 \( 1 - 3.08T + 31T^{2} \)
37 \( 1 - 4.40T + 37T^{2} \)
41 \( 1 + 5.20T + 41T^{2} \)
43 \( 1 + 8.83T + 43T^{2} \)
47 \( 1 + 4.52T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 5.83T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 0.0872T + 71T^{2} \)
73 \( 1 - 9.41T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 5.38T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−8.307431256460753518681917994535, −7.07094723779879153931465189165, −6.49264534609489081614517916689, −6.04004299903550089462023984043, −5.13825627924291544955799469057, −4.83215458504931405364228845544, −3.65815342966886659946228495268, −2.59028313394131813430817073987, −1.66589243005847510347192155768, −1.05002211511824257549310848198, 1.05002211511824257549310848198, 1.66589243005847510347192155768, 2.59028313394131813430817073987, 3.65815342966886659946228495268, 4.83215458504931405364228845544, 5.13825627924291544955799469057, 6.04004299903550089462023984043, 6.49264534609489081614517916689, 7.07094723779879153931465189165, 8.307431256460753518681917994535

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