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  − 1.78·3-s − 0.701·5-s + 2.02·7-s + 0.187·9-s + 0.626·11-s − 2.93·13-s + 1.25·15-s − 2.71·17-s + 1.11·19-s − 3.62·21-s + 0.412·23-s − 4.50·25-s + 5.02·27-s + 6.46·29-s + 4.14·31-s − 1.11·33-s − 1.42·35-s + 2.98·37-s + 5.24·39-s − 0.135·41-s + 0.861·43-s − 0.131·45-s − 1.67·47-s − 2.87·49-s + 4.83·51-s − 8.51·53-s − 0.439·55-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.313·5-s + 0.767·7-s + 0.0624·9-s + 0.189·11-s − 0.814·13-s + 0.323·15-s − 0.657·17-s + 0.254·19-s − 0.790·21-s + 0.0859·23-s − 0.901·25-s + 0.966·27-s + 1.20·29-s + 0.743·31-s − 0.194·33-s − 0.240·35-s + 0.490·37-s + 0.839·39-s − 0.0211·41-s + 0.131·43-s − 0.0195·45-s − 0.244·47-s − 0.411·49-s + 0.677·51-s − 1.16·53-s − 0.0592·55-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 \( 1 + 1.78T + 3T^{2} \)
5 \( 1 + 0.701T + 5T^{2} \)
7 \( 1 - 2.02T + 7T^{2} \)
11 \( 1 - 0.626T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 2.71T + 17T^{2} \)
19 \( 1 - 1.11T + 19T^{2} \)
23 \( 1 - 0.412T + 23T^{2} \)
29 \( 1 - 6.46T + 29T^{2} \)
31 \( 1 - 4.14T + 31T^{2} \)
37 \( 1 - 2.98T + 37T^{2} \)
41 \( 1 + 0.135T + 41T^{2} \)
43 \( 1 - 0.861T + 43T^{2} \)
47 \( 1 + 1.67T + 47T^{2} \)
53 \( 1 + 8.51T + 53T^{2} \)
59 \( 1 - 0.406T + 59T^{2} \)
61 \( 1 + 8.53T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 2.78T + 71T^{2} \)
73 \( 1 + 2.11T + 73T^{2} \)
79 \( 1 - 0.317T + 79T^{2} \)
83 \( 1 - 5.05T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 7.68T + 97T^{2} \)
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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

−7.56653041290657290379467637282, −6.54286363849779373403837562127, −6.25491585586998475988147168703, −5.21603687899626781668021443346, −4.82919088861749603138321487022, −4.18441552387584192800625452641, −3.06429794135468887247100271352, −2.16522293425519912462191933341, −1.05648492980120982710552705863, 0, 1.05648492980120982710552705863, 2.16522293425519912462191933341, 3.06429794135468887247100271352, 4.18441552387584192800625452641, 4.82919088861749603138321487022, 5.21603687899626781668021443346, 6.25491585586998475988147168703, 6.54286363849779373403837562127, 7.56653041290657290379467637282

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