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 − 2.04·5-s + 3.89·7-s + 9-s − 4.70·11-s − 0.326·13-s − 2.04·15-s − 5.01·17-s + 1.82·19-s + 3.89·21-s − 1.20·23-s − 0.807·25-s + 27-s + 2.55·29-s + 7.52·31-s − 4.70·33-s − 7.97·35-s + 9.54·37-s − 0.326·39-s − 8.47·41-s − 4.76·43-s − 2.04·45-s − 4.57·47-s + 8.18·49-s − 5.01·51-s − 1.27·53-s + 9.63·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.915·5-s + 1.47·7-s + 0.333·9-s − 1.41·11-s − 0.0906·13-s − 0.528·15-s − 1.21·17-s + 0.418·19-s + 0.850·21-s − 0.252·23-s − 0.161·25-s + 0.192·27-s + 0.473·29-s + 1.35·31-s − 0.819·33-s − 1.34·35-s + 1.56·37-s − 0.0523·39-s − 1.32·41-s − 0.726·43-s − 0.305·45-s − 0.667·47-s + 1.16·49-s − 0.702·51-s − 0.175·53-s + 1.29·55-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 \( 1 + 2.04T + 5T^{2} \)
7 \( 1 - 3.89T + 7T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 + 0.326T + 13T^{2} \)
17 \( 1 + 5.01T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 - 2.55T + 29T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
37 \( 1 - 9.54T + 37T^{2} \)
41 \( 1 + 8.47T + 41T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + 4.57T + 47T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + 7.43T + 59T^{2} \)
61 \( 1 + 0.825T + 61T^{2} \)
67 \( 1 + 4.57T + 67T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 3.86T + 83T^{2} \)
89 \( 1 + 4.91T + 89T^{2} \)
97 \( 1 + 9.32T + 97T^{2} \)
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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

−7.946641874846197881209169531702, −7.32799440285714355613307035306, −6.42221784491190088250915288712, −5.34051833192675390071259495245, −4.61882617592804616284656579637, −4.28953773677113755553541108624, −3.09527529936488244282532193295, −2.41465203339204784271170965162, −1.43655800615453784533159122833, 0, 1.43655800615453784533159122833, 2.41465203339204784271170965162, 3.09527529936488244282532193295, 4.28953773677113755553541108624, 4.61882617592804616284656579637, 5.34051833192675390071259495245, 6.42221784491190088250915288712, 7.32799440285714355613307035306, 7.946641874846197881209169531702

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