Properties

Degree 2
Conductor $ 2^{4} \cdot 503 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.49·3-s − 1.09·5-s + 0.799·7-s − 0.776·9-s + 0.838·11-s + 2.03·13-s + 1.63·15-s − 1.82·17-s − 4.30·19-s − 1.19·21-s + 0.747·23-s − 3.79·25-s + 5.63·27-s + 2.79·29-s + 0.0226·31-s − 1.25·33-s − 0.877·35-s + 1.80·37-s − 3.03·39-s + 6.90·41-s + 8.84·43-s + 0.852·45-s + 6.50·47-s − 6.36·49-s + 2.71·51-s − 8.42·53-s − 0.920·55-s + ⋯
L(s)  = 1  − 0.860·3-s − 0.490·5-s + 0.302·7-s − 0.258·9-s + 0.252·11-s + 0.563·13-s + 0.422·15-s − 0.442·17-s − 0.987·19-s − 0.260·21-s + 0.155·23-s − 0.759·25-s + 1.08·27-s + 0.518·29-s + 0.00406·31-s − 0.217·33-s − 0.148·35-s + 0.296·37-s − 0.485·39-s + 1.07·41-s + 1.34·43-s + 0.127·45-s + 0.948·47-s − 0.908·49-s + 0.380·51-s − 1.15·53-s − 0.124·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8048\)    =    \(2^{4} \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8048,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + 1.49T + 3T^{2} \)
5 \( 1 + 1.09T + 5T^{2} \)
7 \( 1 - 0.799T + 7T^{2} \)
11 \( 1 - 0.838T + 11T^{2} \)
13 \( 1 - 2.03T + 13T^{2} \)
17 \( 1 + 1.82T + 17T^{2} \)
19 \( 1 + 4.30T + 19T^{2} \)
23 \( 1 - 0.747T + 23T^{2} \)
29 \( 1 - 2.79T + 29T^{2} \)
31 \( 1 - 0.0226T + 31T^{2} \)
37 \( 1 - 1.80T + 37T^{2} \)
41 \( 1 - 6.90T + 41T^{2} \)
43 \( 1 - 8.84T + 43T^{2} \)
47 \( 1 - 6.50T + 47T^{2} \)
53 \( 1 + 8.42T + 53T^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 + 8.08T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 1.18T + 71T^{2} \)
73 \( 1 - 3.90T + 73T^{2} \)
79 \( 1 - 8.22T + 79T^{2} \)
83 \( 1 - 5.12T + 83T^{2} \)
89 \( 1 + 9.57T + 89T^{2} \)
97 \( 1 + 0.843T + 97T^{2} \)
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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

−7.55095820436264941692082324640, −6.55543918451921476210163395361, −6.16058186904638036169520008670, −5.50318006436770672153346388434, −4.55231711705508960618467916077, −4.16690850705804611557406736346, −3.14016788406647018159486109057, −2.18536983472237648809626744480, −1.04124368194136027110486166907, 0, 1.04124368194136027110486166907, 2.18536983472237648809626744480, 3.14016788406647018159486109057, 4.16690850705804611557406736346, 4.55231711705508960618467916077, 5.50318006436770672153346388434, 6.16058186904638036169520008670, 6.55543918451921476210163395361, 7.55095820436264941692082324640

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