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  + 2.25·3-s + 2.34·5-s − 2.23·7-s + 2.09·9-s − 4.83·11-s + 0.834·13-s + 5.28·15-s − 5.54·17-s + 1.83·19-s − 5.04·21-s + 7.33·23-s + 0.483·25-s − 2.05·27-s − 3.35·29-s − 0.777·31-s − 10.9·33-s − 5.23·35-s − 7.78·37-s + 1.88·39-s − 9.17·41-s + 11.5·43-s + 4.89·45-s + 1.92·47-s − 1.99·49-s − 12.5·51-s − 2.04·53-s − 11.3·55-s + ⋯
L(s)  = 1  + 1.30·3-s + 1.04·5-s − 0.845·7-s + 0.696·9-s − 1.45·11-s + 0.231·13-s + 1.36·15-s − 1.34·17-s + 0.420·19-s − 1.10·21-s + 1.52·23-s + 0.0967·25-s − 0.395·27-s − 0.623·29-s − 0.139·31-s − 1.89·33-s − 0.885·35-s − 1.28·37-s + 0.301·39-s − 1.43·41-s + 1.76·43-s + 0.729·45-s + 0.280·47-s − 0.285·49-s − 1.75·51-s − 0.281·53-s − 1.52·55-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 \( 1 - 2.25T + 3T^{2} \)
5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 + 2.23T + 7T^{2} \)
11 \( 1 + 4.83T + 11T^{2} \)
13 \( 1 - 0.834T + 13T^{2} \)
17 \( 1 + 5.54T + 17T^{2} \)
19 \( 1 - 1.83T + 19T^{2} \)
23 \( 1 - 7.33T + 23T^{2} \)
29 \( 1 + 3.35T + 29T^{2} \)
31 \( 1 + 0.777T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + 9.17T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 - 1.92T + 47T^{2} \)
53 \( 1 + 2.04T + 53T^{2} \)
59 \( 1 + 0.668T + 59T^{2} \)
61 \( 1 + 8.64T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 3.57T + 71T^{2} \)
73 \( 1 - 5.69T + 73T^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 8.41T + 89T^{2} \)
97 \( 1 + 3.23T + 97T^{2} \)
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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

−7.42895578203820308081563590936, −6.98473310983976313363314202308, −6.07979225177183923086764393351, −5.44125463946272981404901833262, −4.68388209720231000445452650795, −3.58493720792756095507911954997, −2.93430560824856760204652702376, −2.42335277421438835180683341560, −1.63770770193908761206466838298, 0, 1.63770770193908761206466838298, 2.42335277421438835180683341560, 2.93430560824856760204652702376, 3.58493720792756095507911954997, 4.68388209720231000445452650795, 5.44125463946272981404901833262, 6.07979225177183923086764393351, 6.98473310983976313363314202308, 7.42895578203820308081563590936

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