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 + 2.48·5-s + 1.05·7-s + 9-s − 4.01·11-s − 4.37·13-s + 2.48·15-s − 5.30·17-s + 4.18·19-s + 1.05·21-s − 5.97·23-s + 1.17·25-s + 27-s − 2.50·29-s − 0.980·31-s − 4.01·33-s + 2.62·35-s + 0.119·37-s − 4.37·39-s − 2.22·41-s − 0.812·43-s + 2.48·45-s − 4.81·47-s − 5.88·49-s − 5.30·51-s + 0.831·53-s − 9.98·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.11·5-s + 0.398·7-s + 0.333·9-s − 1.21·11-s − 1.21·13-s + 0.641·15-s − 1.28·17-s + 0.959·19-s + 0.230·21-s − 1.24·23-s + 0.234·25-s + 0.192·27-s − 0.464·29-s − 0.176·31-s − 0.699·33-s + 0.442·35-s + 0.0195·37-s − 0.699·39-s − 0.347·41-s − 0.123·43-s + 0.370·45-s − 0.702·47-s − 0.841·49-s − 0.742·51-s + 0.114·53-s − 1.34·55-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 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 + 4.01T + 11T^{2} \)
13 \( 1 + 4.37T + 13T^{2} \)
17 \( 1 + 5.30T + 17T^{2} \)
19 \( 1 - 4.18T + 19T^{2} \)
23 \( 1 + 5.97T + 23T^{2} \)
29 \( 1 + 2.50T + 29T^{2} \)
31 \( 1 + 0.980T + 31T^{2} \)
37 \( 1 - 0.119T + 37T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 + 0.812T + 43T^{2} \)
47 \( 1 + 4.81T + 47T^{2} \)
53 \( 1 - 0.831T + 53T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 - 3.11T + 61T^{2} \)
67 \( 1 + 2.10T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + 4.21T + 79T^{2} \)
83 \( 1 + 9.78T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.77351190077155118305760158774, −7.12898278694239044157053748184, −6.30540646871648386661301472658, −5.41525104567104614169539108923, −4.99354312090593734058518423081, −4.09847653977046278783064767755, −2.93701740496507162439047018306, −2.27927551005999053936854837649, −1.71568578591468912463258273597, 0, 1.71568578591468912463258273597, 2.27927551005999053936854837649, 2.93701740496507162439047018306, 4.09847653977046278783064767755, 4.99354312090593734058518423081, 5.41525104567104614169539108923, 6.30540646871648386661301472658, 7.12898278694239044157053748184, 7.77351190077155118305760158774

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