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 + 4.20·5-s − 4.85·7-s + 9-s − 2.29·11-s + 4.87·13-s − 4.20·15-s − 5.90·17-s − 6.37·19-s + 4.85·21-s − 7.01·23-s + 12.6·25-s − 27-s − 3.54·29-s + 10.5·31-s + 2.29·33-s − 20.4·35-s − 6.92·37-s − 4.87·39-s + 7.28·41-s − 0.286·43-s + 4.20·45-s + 10.1·47-s + 16.5·49-s + 5.90·51-s − 5.30·53-s − 9.64·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.87·5-s − 1.83·7-s + 0.333·9-s − 0.691·11-s + 1.35·13-s − 1.08·15-s − 1.43·17-s − 1.46·19-s + 1.05·21-s − 1.46·23-s + 2.53·25-s − 0.192·27-s − 0.658·29-s + 1.89·31-s + 0.399·33-s − 3.44·35-s − 1.13·37-s − 0.780·39-s + 1.13·41-s − 0.0436·43-s + 0.626·45-s + 1.48·47-s + 2.36·49-s + 0.827·51-s − 0.728·53-s − 1.30·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.472551031$
$L(\frac12)$  $\approx$  $1.472551031$
$L(\frac{3}{2})$   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 \( 1 - 4.20T + 5T^{2} \)
7 \( 1 + 4.85T + 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 - 4.87T + 13T^{2} \)
17 \( 1 + 5.90T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 + 7.01T + 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 - 7.28T + 41T^{2} \)
43 \( 1 + 0.286T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 5.30T + 53T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 - 5.18T + 67T^{2} \)
71 \( 1 - 6.89T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 - 8.84T + 79T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−8.280025827134800526961304130499, −6.84187425426738381568752260329, −6.47259847715901976241241429446, −6.05062517044723763322247169672, −5.59663657278735430275758226252, −4.47894195034286469927088868682, −3.65127221444219421631499416303, −2.49801089363823363473684003559, −2.06335035160172049393690044832, −0.62608620390795535423288063813, 0.62608620390795535423288063813, 2.06335035160172049393690044832, 2.49801089363823363473684003559, 3.65127221444219421631499416303, 4.47894195034286469927088868682, 5.59663657278735430275758226252, 6.05062517044723763322247169672, 6.47259847715901976241241429446, 6.84187425426738381568752260329, 8.280025827134800526961304130499

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