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 − 0.642·5-s − 2.57·7-s + 9-s − 1.59·11-s − 3.01·13-s − 0.642·15-s + 5.85·17-s + 4.15·19-s − 2.57·21-s + 6.16·23-s − 4.58·25-s + 27-s + 0.244·29-s − 5.80·31-s − 1.59·33-s + 1.65·35-s − 2.35·37-s − 3.01·39-s + 3.00·41-s − 1.02·43-s − 0.642·45-s − 9.29·47-s − 0.359·49-s + 5.85·51-s − 8.77·53-s + 1.02·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.287·5-s − 0.973·7-s + 0.333·9-s − 0.480·11-s − 0.837·13-s − 0.165·15-s + 1.41·17-s + 0.953·19-s − 0.562·21-s + 1.28·23-s − 0.917·25-s + 0.192·27-s + 0.0454·29-s − 1.04·31-s − 0.277·33-s + 0.279·35-s − 0.387·37-s − 0.483·39-s + 0.469·41-s − 0.155·43-s − 0.0957·45-s − 1.35·47-s − 0.0513·49-s + 0.819·51-s − 1.20·53-s + 0.138·55-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 \( 1 + 0.642T + 5T^{2} \)
7 \( 1 + 2.57T + 7T^{2} \)
11 \( 1 + 1.59T + 11T^{2} \)
13 \( 1 + 3.01T + 13T^{2} \)
17 \( 1 - 5.85T + 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
23 \( 1 - 6.16T + 23T^{2} \)
29 \( 1 - 0.244T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + 2.35T + 37T^{2} \)
41 \( 1 - 3.00T + 41T^{2} \)
43 \( 1 + 1.02T + 43T^{2} \)
47 \( 1 + 9.29T + 47T^{2} \)
53 \( 1 + 8.77T + 53T^{2} \)
59 \( 1 - 1.90T + 59T^{2} \)
61 \( 1 - 3.09T + 61T^{2} \)
67 \( 1 - 0.795T + 67T^{2} \)
71 \( 1 - 6.62T + 71T^{2} \)
73 \( 1 - 5.75T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 5.53T + 83T^{2} \)
89 \( 1 - 6.54T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−7.66756315554763890959941899467, −7.21685131662174269055977440048, −6.42140632592927415815028034443, −5.43216411640942093647142641148, −4.95896338798984186334293017285, −3.72937600399808326029392644728, −3.25781804993775580954388633411, −2.56344323193414432664447243098, −1.33199506475897315149074044766, 0, 1.33199506475897315149074044766, 2.56344323193414432664447243098, 3.25781804993775580954388633411, 3.72937600399808326029392644728, 4.95896338798984186334293017285, 5.43216411640942093647142641148, 6.42140632592927415815028034443, 7.21685131662174269055977440048, 7.66756315554763890959941899467

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