Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.61·5-s + 0.982·7-s + 9-s − 3.24·11-s + 2.88·13-s − 3.61·15-s − 0.659·17-s + 2.91·19-s + 0.982·21-s − 1.79·23-s + 8.03·25-s + 27-s + 4.94·29-s + 1.83·31-s − 3.24·33-s − 3.54·35-s − 11.7·37-s + 2.88·39-s − 3.21·41-s − 7.99·43-s − 3.61·45-s + 6.45·47-s − 6.03·49-s − 0.659·51-s + 1.47·53-s + 11.7·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.61·5-s + 0.371·7-s + 0.333·9-s − 0.978·11-s + 0.800·13-s − 0.932·15-s − 0.160·17-s + 0.668·19-s + 0.214·21-s − 0.374·23-s + 1.60·25-s + 0.192·27-s + 0.918·29-s + 0.330·31-s − 0.564·33-s − 0.599·35-s − 1.93·37-s + 0.462·39-s − 0.501·41-s − 1.21·43-s − 0.538·45-s + 0.941·47-s − 0.862·49-s − 0.0923·51-s + 0.202·53-s + 1.57·55-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  =  1
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + 3.61T + 5T^{2} \)
7 \( 1 - 0.982T + 7T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 + 0.659T + 17T^{2} \)
19 \( 1 - 2.91T + 19T^{2} \)
23 \( 1 + 1.79T + 23T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 - 1.83T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + 3.21T + 41T^{2} \)
43 \( 1 + 7.99T + 43T^{2} \)
47 \( 1 - 6.45T + 47T^{2} \)
53 \( 1 - 1.47T + 53T^{2} \)
59 \( 1 + 0.809T + 59T^{2} \)
61 \( 1 - 9.89T + 61T^{2} \)
67 \( 1 - 7.49T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 - 3.32T + 83T^{2} \)
89 \( 1 + 7.58T + 89T^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−7.894152508355267000821939494809, −7.18833626352020600292131824784, −6.56658264785904825767766857634, −5.32472165421513135289404596556, −4.77693043082299225068255602703, −3.83593856507465345901427494134, −3.40626078466208069477809746736, −2.49764327590567590828342408731, −1.26008824744431192658065527817, 0, 1.26008824744431192658065527817, 2.49764327590567590828342408731, 3.40626078466208069477809746736, 3.83593856507465345901427494134, 4.77693043082299225068255602703, 5.32472165421513135289404596556, 6.56658264785904825767766857634, 7.18833626352020600292131824784, 7.894152508355267000821939494809

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