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 − 2.74·5-s + 1.46·7-s + 9-s + 4.30·11-s + 3.43·13-s + 2.74·15-s + 7.91·17-s + 1.80·19-s − 1.46·21-s + 1.83·23-s + 2.56·25-s − 27-s + 8.64·29-s + 3.70·31-s − 4.30·33-s − 4.03·35-s + 11.8·37-s − 3.43·39-s + 0.510·41-s − 7.96·43-s − 2.74·45-s − 4.97·47-s − 4.84·49-s − 7.91·51-s − 13.7·53-s − 11.8·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.22·5-s + 0.554·7-s + 0.333·9-s + 1.29·11-s + 0.951·13-s + 0.709·15-s + 1.91·17-s + 0.413·19-s − 0.320·21-s + 0.383·23-s + 0.512·25-s − 0.192·27-s + 1.60·29-s + 0.665·31-s − 0.750·33-s − 0.681·35-s + 1.94·37-s − 0.549·39-s + 0.0797·41-s − 1.21·43-s − 0.409·45-s − 0.725·47-s − 0.692·49-s − 1.10·51-s − 1.88·53-s − 1.59·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.870285808$
$L(\frac12)$  $\approx$  $1.870285808$
$L(\frac{3}{2})$   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 \( 1 + 2.74T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
13 \( 1 - 3.43T + 13T^{2} \)
17 \( 1 - 7.91T + 17T^{2} \)
19 \( 1 - 1.80T + 19T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 - 8.64T + 29T^{2} \)
31 \( 1 - 3.70T + 31T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 - 0.510T + 41T^{2} \)
43 \( 1 + 7.96T + 43T^{2} \)
47 \( 1 + 4.97T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
61 \( 1 + 0.965T + 61T^{2} \)
67 \( 1 - 9.78T + 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 8.61T + 73T^{2} \)
79 \( 1 - 4.80T + 79T^{2} \)
83 \( 1 - 9.61T + 83T^{2} \)
89 \( 1 - 6.41T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−8.009853707865882964359991875045, −7.53047401927513677008386763540, −6.49832683525863111935873037343, −6.15283635527596841830307736394, −5.03204219584919781127558751008, −4.49515990587268019130454916412, −3.64846211413304111779088884841, −3.12326105849475193709522395915, −1.38951692105109310090383397631, −0.876292476308589115466420585488, 0.876292476308589115466420585488, 1.38951692105109310090383397631, 3.12326105849475193709522395915, 3.64846211413304111779088884841, 4.49515990587268019130454916412, 5.03204219584919781127558751008, 6.15283635527596841830307736394, 6.49832683525863111935873037343, 7.53047401927513677008386763540, 8.009853707865882964359991875045

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