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  − 2.25·5-s − 4.66·7-s + 0.681·11-s + 0.464·13-s + 0.986·17-s − 2.32·19-s + 5.37·23-s + 0.0786·25-s + 4.75·29-s − 1.57·31-s + 10.5·35-s + 3.55·37-s − 0.278·41-s − 1.78·43-s − 2.22·47-s + 14.8·49-s − 2.88·53-s − 1.53·55-s + 6.32·59-s + 0.0560·61-s − 1.04·65-s − 5.83·67-s + 2.74·71-s + 9.63·73-s − 3.18·77-s + 3.00·79-s + 2.32·83-s + ⋯
L(s)  = 1  − 1.00·5-s − 1.76·7-s + 0.205·11-s + 0.128·13-s + 0.239·17-s − 0.534·19-s + 1.12·23-s + 0.0157·25-s + 0.882·29-s − 0.282·31-s + 1.77·35-s + 0.583·37-s − 0.0434·41-s − 0.271·43-s − 0.324·47-s + 2.11·49-s − 0.396·53-s − 0.207·55-s + 0.823·59-s + 0.00717·61-s − 0.129·65-s − 0.713·67-s + 0.325·71-s + 1.12·73-s − 0.362·77-s + 0.338·79-s + 0.255·83-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 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 - 0.681T + 11T^{2} \)
13 \( 1 - 0.464T + 13T^{2} \)
17 \( 1 - 0.986T + 17T^{2} \)
19 \( 1 + 2.32T + 19T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 - 4.75T + 29T^{2} \)
31 \( 1 + 1.57T + 31T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 + 0.278T + 41T^{2} \)
43 \( 1 + 1.78T + 43T^{2} \)
47 \( 1 + 2.22T + 47T^{2} \)
53 \( 1 + 2.88T + 53T^{2} \)
59 \( 1 - 6.32T + 59T^{2} \)
61 \( 1 - 0.0560T + 61T^{2} \)
67 \( 1 + 5.83T + 67T^{2} \)
71 \( 1 - 2.74T + 71T^{2} \)
73 \( 1 - 9.63T + 73T^{2} \)
79 \( 1 - 3.00T + 79T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 4.01T + 97T^{2} \)
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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

−7.35119531015346293482627794501, −6.61594958757646763682491813235, −6.30444404542713079411161552617, −5.33033482173625336758491361709, −4.45079754613014084772607129514, −3.68332892679549857685004084236, −3.24773089520048471410481789320, −2.42207729277841433762271671245, −0.945364539623739597008874214982, 0, 0.945364539623739597008874214982, 2.42207729277841433762271671245, 3.24773089520048471410481789320, 3.68332892679549857685004084236, 4.45079754613014084772607129514, 5.33033482173625336758491361709, 6.30444404542713079411161552617, 6.61594958757646763682491813235, 7.35119531015346293482627794501

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