Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 251 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.454·5-s − 2.18·7-s + 1.15·11-s + 6.03·13-s − 0.0535·17-s − 2.08·19-s − 3.22·23-s − 4.79·25-s + 1.62·29-s − 2.01·31-s + 0.993·35-s − 3.49·37-s − 2.85·41-s + 2.78·43-s + 9.03·47-s − 2.21·49-s + 4.09·53-s − 0.526·55-s + 4.76·59-s − 14.6·61-s − 2.74·65-s − 4.98·67-s + 2.04·71-s − 0.0937·73-s − 2.53·77-s − 1.92·79-s − 7.96·83-s + ⋯
L(s)  = 1  − 0.203·5-s − 0.826·7-s + 0.349·11-s + 1.67·13-s − 0.0129·17-s − 0.479·19-s − 0.673·23-s − 0.958·25-s + 0.300·29-s − 0.362·31-s + 0.167·35-s − 0.574·37-s − 0.446·41-s + 0.423·43-s + 1.31·47-s − 0.317·49-s + 0.561·53-s − 0.0710·55-s + 0.620·59-s − 1.87·61-s − 0.340·65-s − 0.609·67-s + 0.242·71-s − 0.0109·73-s − 0.288·77-s − 0.216·79-s − 0.874·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9036\)    =    \(2^{2} \cdot 3^{2} \cdot 251\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;251\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
251 \( 1 + T \)
good5 \( 1 + 0.454T + 5T^{2} \)
7 \( 1 + 2.18T + 7T^{2} \)
11 \( 1 - 1.15T + 11T^{2} \)
13 \( 1 - 6.03T + 13T^{2} \)
17 \( 1 + 0.0535T + 17T^{2} \)
19 \( 1 + 2.08T + 19T^{2} \)
23 \( 1 + 3.22T + 23T^{2} \)
29 \( 1 - 1.62T + 29T^{2} \)
31 \( 1 + 2.01T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 - 2.78T + 43T^{2} \)
47 \( 1 - 9.03T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 4.98T + 67T^{2} \)
71 \( 1 - 2.04T + 71T^{2} \)
73 \( 1 + 0.0937T + 73T^{2} \)
79 \( 1 + 1.92T + 79T^{2} \)
83 \( 1 + 7.96T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 - 3.00T + 97T^{2} \)
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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

−7.39998199104708574258525399949, −6.54269970359087954925261831863, −6.12915897333396176329049174668, −5.53193482275613286110424105139, −4.37722237699094667308227741076, −3.79649144760933204555594517155, −3.25360386900404939876376984563, −2.16668427083624228992803294578, −1.21095353457914458271724858244, 0, 1.21095353457914458271724858244, 2.16668427083624228992803294578, 3.25360386900404939876376984563, 3.79649144760933204555594517155, 4.37722237699094667308227741076, 5.53193482275613286110424105139, 6.12915897333396176329049174668, 6.54269970359087954925261831863, 7.39998199104708574258525399949

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