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 + 2.64·5-s − 3.02·7-s + 9-s + 0.587·11-s − 2.23·13-s + 2.64·15-s + 1.24·17-s − 0.491·19-s − 3.02·21-s − 0.916·23-s + 1.99·25-s + 27-s − 5.33·29-s − 8.59·31-s + 0.587·33-s − 7.99·35-s − 11.8·37-s − 2.23·39-s − 2.54·41-s + 1.12·43-s + 2.64·45-s − 2.52·47-s + 2.13·49-s + 1.24·51-s − 0.789·53-s + 1.55·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.18·5-s − 1.14·7-s + 0.333·9-s + 0.177·11-s − 0.619·13-s + 0.682·15-s + 0.302·17-s − 0.112·19-s − 0.659·21-s − 0.191·23-s + 0.398·25-s + 0.192·27-s − 0.990·29-s − 1.54·31-s + 0.102·33-s − 1.35·35-s − 1.95·37-s − 0.357·39-s − 0.397·41-s + 0.170·43-s + 0.394·45-s − 0.368·47-s + 0.304·49-s + 0.174·51-s − 0.108·53-s + 0.209·55-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 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 + 3.02T + 7T^{2} \)
11 \( 1 - 0.587T + 11T^{2} \)
13 \( 1 + 2.23T + 13T^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
19 \( 1 + 0.491T + 19T^{2} \)
23 \( 1 + 0.916T + 23T^{2} \)
29 \( 1 + 5.33T + 29T^{2} \)
31 \( 1 + 8.59T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + 2.52T + 47T^{2} \)
53 \( 1 + 0.789T + 53T^{2} \)
59 \( 1 - 9.14T + 59T^{2} \)
61 \( 1 + 9.78T + 61T^{2} \)
67 \( 1 + 7.68T + 67T^{2} \)
71 \( 1 - 3.65T + 71T^{2} \)
73 \( 1 + 6.58T + 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 3.28T + 89T^{2} \)
97 \( 1 + 8.50T + 97T^{2} \)
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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

−7.58447329253587984453841347909, −7.02478097943508157309177764350, −6.30830331056909325604019152821, −5.64575996775182940662538624228, −4.97112621655505460656044140737, −3.76531128441178261733027798650, −3.24594727472379110715074391868, −2.25739287819806364572051122350, −1.63515291823377262767417812054, 0, 1.63515291823377262767417812054, 2.25739287819806364572051122350, 3.24594727472379110715074391868, 3.76531128441178261733027798650, 4.97112621655505460656044140737, 5.64575996775182940662538624228, 6.30830331056909325604019152821, 7.02478097943508157309177764350, 7.58447329253587984453841347909

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