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  + 2.47·5-s − 0.972·7-s − 3.50·11-s + 2.87·13-s + 2.27·17-s − 6.78·19-s − 3.58·23-s + 1.10·25-s + 7.60·29-s − 0.279·31-s − 2.40·35-s − 9.37·37-s + 10.4·41-s − 5.70·43-s + 8.12·47-s − 6.05·49-s − 1.01·53-s − 8.66·55-s − 12.3·59-s + 9.95·61-s + 7.11·65-s − 4.50·67-s − 9.82·71-s + 6.37·73-s + 3.41·77-s + 1.13·79-s + 1.53·83-s + ⋯
L(s)  = 1  + 1.10·5-s − 0.367·7-s − 1.05·11-s + 0.798·13-s + 0.550·17-s − 1.55·19-s − 0.746·23-s + 0.220·25-s + 1.41·29-s − 0.0502·31-s − 0.405·35-s − 1.54·37-s + 1.63·41-s − 0.869·43-s + 1.18·47-s − 0.864·49-s − 0.139·53-s − 1.16·55-s − 1.60·59-s + 1.27·61-s + 0.882·65-s − 0.550·67-s − 1.16·71-s + 0.745·73-s + 0.388·77-s + 0.127·79-s + 0.168·83-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 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 + 0.972T + 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 - 2.27T + 17T^{2} \)
19 \( 1 + 6.78T + 19T^{2} \)
23 \( 1 + 3.58T + 23T^{2} \)
29 \( 1 - 7.60T + 29T^{2} \)
31 \( 1 + 0.279T + 31T^{2} \)
37 \( 1 + 9.37T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 - 8.12T + 47T^{2} \)
53 \( 1 + 1.01T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 9.95T + 61T^{2} \)
67 \( 1 + 4.50T + 67T^{2} \)
71 \( 1 + 9.82T + 71T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 - 1.53T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + 0.254T + 97T^{2} \)
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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

−7.35624241056723137913962974876, −6.49415697003979755344727099305, −6.05210050721856523174900382873, −5.49063588726123055737670348069, −4.66701169081729190967107831828, −3.83865434738355801793398512679, −2.88784302235533235207541194747, −2.23340574366760145010237850814, −1.36945941745934456908033236700, 0, 1.36945941745934456908033236700, 2.23340574366760145010237850814, 2.88784302235533235207541194747, 3.83865434738355801793398512679, 4.66701169081729190967107831828, 5.49063588726123055737670348069, 6.05210050721856523174900382873, 6.49415697003979755344727099305, 7.35624241056723137913962974876

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