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 + 1.66·5-s − 2.19·7-s + 9-s − 5.33·11-s + 2.06·13-s − 1.66·15-s + 3.14·17-s − 6.10·19-s + 2.19·21-s + 6.59·23-s − 2.24·25-s − 27-s + 8.96·29-s − 6.89·31-s + 5.33·33-s − 3.63·35-s − 4.91·37-s − 2.06·39-s + 5.27·41-s + 4.91·43-s + 1.66·45-s + 0.585·47-s − 2.19·49-s − 3.14·51-s − 9.07·53-s − 8.86·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.742·5-s − 0.828·7-s + 0.333·9-s − 1.60·11-s + 0.573·13-s − 0.428·15-s + 0.762·17-s − 1.40·19-s + 0.478·21-s + 1.37·23-s − 0.448·25-s − 0.192·27-s + 1.66·29-s − 1.23·31-s + 0.929·33-s − 0.615·35-s − 0.808·37-s − 0.330·39-s + 0.824·41-s + 0.749·43-s + 0.247·45-s + 0.0853·47-s − 0.313·49-s − 0.440·51-s − 1.24·53-s − 1.19·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.236690592$
$L(\frac12)$  $\approx$  $1.236690592$
$L(\frac{3}{2})$   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 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
11 \( 1 + 5.33T + 11T^{2} \)
13 \( 1 - 2.06T + 13T^{2} \)
17 \( 1 - 3.14T + 17T^{2} \)
19 \( 1 + 6.10T + 19T^{2} \)
23 \( 1 - 6.59T + 23T^{2} \)
29 \( 1 - 8.96T + 29T^{2} \)
31 \( 1 + 6.89T + 31T^{2} \)
37 \( 1 + 4.91T + 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 - 4.91T + 43T^{2} \)
47 \( 1 - 0.585T + 47T^{2} \)
53 \( 1 + 9.07T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 6.61T + 61T^{2} \)
67 \( 1 + 5.16T + 67T^{2} \)
71 \( 1 - 0.933T + 71T^{2} \)
73 \( 1 - 6.11T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 5.02T + 89T^{2} \)
97 \( 1 - 6.45T + 97T^{2} \)
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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

−8.005879650024517577679126721595, −7.32176147764711308321618509134, −6.38825119165555778111468500239, −6.07262206963247613312314327389, −5.25490365184010225363452680227, −4.69398204442922015164775962788, −3.52042913213452135722164150778, −2.78250061269915479770506960703, −1.86693765633724989809397531691, −0.58491676842469093239322489245, 0.58491676842469093239322489245, 1.86693765633724989809397531691, 2.78250061269915479770506960703, 3.52042913213452135722164150778, 4.69398204442922015164775962788, 5.25490365184010225363452680227, 6.07262206963247613312314327389, 6.38825119165555778111468500239, 7.32176147764711308321618509134, 8.005879650024517577679126721595

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