Properties

Degree 2
Conductor $ 2 \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  − 2-s + 4-s − 8-s − 3·9-s + 4·11-s + 2·13-s + 16-s − 2·17-s + 3·18-s − 2·19-s − 4·22-s + 8·23-s − 5·25-s − 2·26-s + 8·31-s − 32-s + 2·34-s − 3·36-s + 8·37-s + 2·38-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 8·47-s − 7·49-s + 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s − 0.852·22-s + 1.66·23-s − 25-s − 0.392·26-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.31·37-s + 0.324·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 1.16·47-s − 49-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1006\)    =    \(2 \cdot 503\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1006} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1006,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.138501340$
$L(\frac12)$  $\approx$  $1.138501340$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
503 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.90119148272826, −19.46218211683029, −18.98576445062456, −17.81025207011486, −17.53077035407458, −16.77702619042166, −16.23022669698980, −15.16921414184883, −14.74232263226916, −13.82773901692190, −13.08254116975267, −12.01380616176093, −11.36955148933910, −10.91035490665348, −9.776853377242167, −9.045713520834134, −8.518177580053213, −7.601497559220954, −6.481299407931071, −6.053108181685923, −4.690494471727124, −3.517824015761788, −2.398920434570393, −0.9462573344587738, 0.9462573344587738, 2.398920434570393, 3.517824015761788, 4.690494471727124, 6.053108181685923, 6.481299407931071, 7.601497559220954, 8.518177580053213, 9.045713520834134, 9.776853377242167, 10.91035490665348, 11.36955148933910, 12.01380616176093, 13.08254116975267, 13.82773901692190, 14.74232263226916, 15.16921414184883, 16.23022669698980, 16.77702619042166, 17.53077035407458, 17.81025207011486, 18.98576445062456, 19.46218211683029, 19.90119148272826

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