Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.00·5-s + 4.42·7-s + 9-s − 1.26·11-s − 7.02·13-s + 3.00·15-s + 4.46·17-s − 6.47·19-s − 4.42·21-s − 5.72·23-s + 4.04·25-s − 27-s − 5.47·29-s + 3.83·31-s + 1.26·33-s − 13.3·35-s + 1.66·37-s + 7.02·39-s + 3.44·41-s + 7.06·43-s − 3.00·45-s + 2.66·47-s + 12.5·49-s − 4.46·51-s − 11.5·53-s + 3.79·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 1.67·7-s + 0.333·9-s − 0.380·11-s − 1.94·13-s + 0.776·15-s + 1.08·17-s − 1.48·19-s − 0.965·21-s − 1.19·23-s + 0.808·25-s − 0.192·27-s − 1.01·29-s + 0.687·31-s + 0.219·33-s − 2.24·35-s + 0.274·37-s + 1.12·39-s + 0.538·41-s + 1.07·43-s − 0.448·45-s + 0.388·47-s + 1.79·49-s − 0.625·51-s − 1.58·53-s + 0.511·55-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  =  0
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8508215625$
$L(\frac12)$  $\approx$  $0.8508215625$
$L(\frac{3}{2})$   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 \( 1 + 3.00T + 5T^{2} \)
7 \( 1 - 4.42T + 7T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + 7.02T + 13T^{2} \)
17 \( 1 - 4.46T + 17T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 + 5.72T + 23T^{2} \)
29 \( 1 + 5.47T + 29T^{2} \)
31 \( 1 - 3.83T + 31T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 - 3.44T + 41T^{2} \)
43 \( 1 - 7.06T + 43T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 8.23T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 7.21T + 67T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + 5.90T + 79T^{2} \)
83 \( 1 - 2.99T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 9.56T + 97T^{2} \)
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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

−7.974925935330775927725964787199, −7.55404977772972906046212316126, −6.90293156584032551939407351109, −5.69254582856676851595706442169, −5.16793898148002160130131372872, −4.27925277160459172480525317097, −4.14744656494423976745085298384, −2.65862304203081840648378457415, −1.83070813155896868398567591001, −0.48860877890919594995270846709, 0.48860877890919594995270846709, 1.83070813155896868398567591001, 2.65862304203081840648378457415, 4.14744656494423976745085298384, 4.27925277160459172480525317097, 5.16793898148002160130131372872, 5.69254582856676851595706442169, 6.90293156584032551939407351109, 7.55404977772972906046212316126, 7.974925935330775927725964787199

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