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  + 0.667·3-s + 0.307·5-s − 4.25·7-s − 2.55·9-s − 0.721·11-s − 4.50·13-s + 0.204·15-s + 7.96·17-s + 6.29·19-s − 2.83·21-s + 4.88·23-s − 4.90·25-s − 3.70·27-s + 7.38·29-s + 8.57·31-s − 0.481·33-s − 1.30·35-s − 0.903·37-s − 3.00·39-s − 9.48·41-s + 1.86·43-s − 0.784·45-s + 0.432·47-s + 11.0·49-s + 5.31·51-s − 13.1·53-s − 0.221·55-s + ⋯
L(s)  = 1  + 0.385·3-s + 0.137·5-s − 1.60·7-s − 0.851·9-s − 0.217·11-s − 1.25·13-s + 0.0529·15-s + 1.93·17-s + 1.44·19-s − 0.618·21-s + 1.01·23-s − 0.981·25-s − 0.713·27-s + 1.37·29-s + 1.53·31-s − 0.0837·33-s − 0.220·35-s − 0.148·37-s − 0.481·39-s − 1.48·41-s + 0.283·43-s − 0.116·45-s + 0.0631·47-s + 1.58·49-s + 0.744·51-s − 1.80·53-s − 0.0298·55-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 \( 1 - 0.667T + 3T^{2} \)
5 \( 1 - 0.307T + 5T^{2} \)
7 \( 1 + 4.25T + 7T^{2} \)
11 \( 1 + 0.721T + 11T^{2} \)
13 \( 1 + 4.50T + 13T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
19 \( 1 - 6.29T + 19T^{2} \)
23 \( 1 - 4.88T + 23T^{2} \)
29 \( 1 - 7.38T + 29T^{2} \)
31 \( 1 - 8.57T + 31T^{2} \)
37 \( 1 + 0.903T + 37T^{2} \)
41 \( 1 + 9.48T + 41T^{2} \)
43 \( 1 - 1.86T + 43T^{2} \)
47 \( 1 - 0.432T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 3.80T + 59T^{2} \)
61 \( 1 + 6.98T + 61T^{2} \)
67 \( 1 + 4.59T + 67T^{2} \)
71 \( 1 + 1.98T + 71T^{2} \)
73 \( 1 - 3.07T + 73T^{2} \)
79 \( 1 + 1.39T + 79T^{2} \)
83 \( 1 - 8.69T + 83T^{2} \)
89 \( 1 - 3.04T + 89T^{2} \)
97 \( 1 + 5.67T + 97T^{2} \)
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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

−7.68026510938078090119921050023, −6.74306933719791183956611373636, −6.15175149318414974576392372350, −5.39255738933506810878863003793, −4.84141229490105014301087632971, −3.49033143971759427082820711348, −3.05725533991091081439275432343, −2.65457374092320846577023622419, −1.15155196753552591261035668284, 0, 1.15155196753552591261035668284, 2.65457374092320846577023622419, 3.05725533991091081439275432343, 3.49033143971759427082820711348, 4.84141229490105014301087632971, 5.39255738933506810878863003793, 6.15175149318414974576392372350, 6.74306933719791183956611373636, 7.68026510938078090119921050023

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