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 − 0.453·5-s + 2.17·7-s + 9-s − 5.35·11-s + 2.71·13-s − 0.453·15-s + 6.07·17-s − 2.86·19-s + 2.17·21-s − 5.42·23-s − 4.79·25-s + 27-s − 7.45·29-s − 5.78·31-s − 5.35·33-s − 0.983·35-s + 1.10·37-s + 2.71·39-s − 6.10·41-s + 6.08·43-s − 0.453·45-s − 3.32·47-s − 2.28·49-s + 6.07·51-s − 5.31·53-s + 2.42·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.202·5-s + 0.820·7-s + 0.333·9-s − 1.61·11-s + 0.752·13-s − 0.116·15-s + 1.47·17-s − 0.656·19-s + 0.473·21-s − 1.13·23-s − 0.958·25-s + 0.192·27-s − 1.38·29-s − 1.03·31-s − 0.932·33-s − 0.166·35-s + 0.181·37-s + 0.434·39-s − 0.953·41-s + 0.928·43-s − 0.0675·45-s − 0.485·47-s − 0.326·49-s + 0.851·51-s − 0.730·53-s + 0.327·55-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 \( 1 + 0.453T + 5T^{2} \)
7 \( 1 - 2.17T + 7T^{2} \)
11 \( 1 + 5.35T + 11T^{2} \)
13 \( 1 - 2.71T + 13T^{2} \)
17 \( 1 - 6.07T + 17T^{2} \)
19 \( 1 + 2.86T + 19T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 + 7.45T + 29T^{2} \)
31 \( 1 + 5.78T + 31T^{2} \)
37 \( 1 - 1.10T + 37T^{2} \)
41 \( 1 + 6.10T + 41T^{2} \)
43 \( 1 - 6.08T + 43T^{2} \)
47 \( 1 + 3.32T + 47T^{2} \)
53 \( 1 + 5.31T + 53T^{2} \)
59 \( 1 - 1.97T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 3.24T + 67T^{2} \)
71 \( 1 - 3.70T + 71T^{2} \)
73 \( 1 - 2.73T + 73T^{2} \)
79 \( 1 - 2.25T + 79T^{2} \)
83 \( 1 - 0.0721T + 83T^{2} \)
89 \( 1 + 9.49T + 89T^{2} \)
97 \( 1 + 2.29T + 97T^{2} \)
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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

−7.80071419831304046704618091258, −7.45587282466604815812946109584, −6.14911670845746861792254212265, −5.55865805973858160831044431510, −4.85844311616059545467650321779, −3.88983009174014842587999636367, −3.32014652907163737874493238281, −2.22942785802605598597860674717, −1.56436602121338502018088697102, 0, 1.56436602121338502018088697102, 2.22942785802605598597860674717, 3.32014652907163737874493238281, 3.88983009174014842587999636367, 4.85844311616059545467650321779, 5.55865805973858160831044431510, 6.14911670845746861792254212265, 7.45587282466604815812946109584, 7.80071419831304046704618091258

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