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 + 3.20·5-s − 2.04·7-s + 9-s + 0.355·11-s − 1.61·13-s − 3.20·15-s − 1.55·17-s + 4.81·19-s + 2.04·21-s − 1.33·23-s + 5.28·25-s − 27-s + 7.93·29-s + 0.483·31-s − 0.355·33-s − 6.55·35-s + 1.13·37-s + 1.61·39-s + 2.30·41-s − 8.90·43-s + 3.20·45-s − 8.25·47-s − 2.81·49-s + 1.55·51-s + 5.66·53-s + 1.13·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.43·5-s − 0.773·7-s + 0.333·9-s + 0.107·11-s − 0.447·13-s − 0.827·15-s − 0.376·17-s + 1.10·19-s + 0.446·21-s − 0.278·23-s + 1.05·25-s − 0.192·27-s + 1.47·29-s + 0.0868·31-s − 0.0618·33-s − 1.10·35-s + 0.187·37-s + 0.258·39-s + 0.360·41-s − 1.35·43-s + 0.478·45-s − 1.20·47-s − 0.402·49-s + 0.217·51-s + 0.777·53-s + 0.153·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.923327071$
$L(\frac12)$  $\approx$  $1.923327071$
$L(\frac{3}{2})$   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 \( 1 - 3.20T + 5T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 - 0.355T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 - 4.81T + 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 - 0.483T + 31T^{2} \)
37 \( 1 - 1.13T + 37T^{2} \)
41 \( 1 - 2.30T + 41T^{2} \)
43 \( 1 + 8.90T + 43T^{2} \)
47 \( 1 + 8.25T + 47T^{2} \)
53 \( 1 - 5.66T + 53T^{2} \)
59 \( 1 - 8.23T + 59T^{2} \)
61 \( 1 - 9.44T + 61T^{2} \)
67 \( 1 - 3.12T + 67T^{2} \)
71 \( 1 - 2.32T + 71T^{2} \)
73 \( 1 + 5.59T + 73T^{2} \)
79 \( 1 - 1.77T + 79T^{2} \)
83 \( 1 - 1.56T + 83T^{2} \)
89 \( 1 - 8.47T + 89T^{2} \)
97 \( 1 - 19.3T + 97T^{2} \)
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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

−8.092460433262617273780269283965, −7.05680281853266114763943235327, −6.55446401561524259072281948899, −6.01798551668109598764121604966, −5.26601877203548109774198129641, −4.73261844009382130668516395186, −3.54722211793180244273023232096, −2.69136299628662306498366466685, −1.83929932821646630391721846243, −0.75485184158611352864356976433, 0.75485184158611352864356976433, 1.83929932821646630391721846243, 2.69136299628662306498366466685, 3.54722211793180244273023232096, 4.73261844009382130668516395186, 5.26601877203548109774198129641, 6.01798551668109598764121604966, 6.55446401561524259072281948899, 7.05680281853266114763943235327, 8.092460433262617273780269283965

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