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  − 2.31·3-s + 0.927·5-s + 1.63·7-s + 2.34·9-s − 2.24·11-s − 4.71·13-s − 2.14·15-s − 2.14·17-s + 6.96·19-s − 3.78·21-s + 5.04·23-s − 4.13·25-s + 1.52·27-s − 5.17·29-s − 4.00·31-s + 5.18·33-s + 1.51·35-s − 0.244·37-s + 10.9·39-s + 7.08·41-s + 9.58·43-s + 2.17·45-s + 5.81·47-s − 4.31·49-s + 4.94·51-s + 2.15·53-s − 2.07·55-s + ⋯
L(s)  = 1  − 1.33·3-s + 0.414·5-s + 0.619·7-s + 0.780·9-s − 0.675·11-s − 1.30·13-s − 0.553·15-s − 0.519·17-s + 1.59·19-s − 0.826·21-s + 1.05·23-s − 0.827·25-s + 0.292·27-s − 0.960·29-s − 0.718·31-s + 0.901·33-s + 0.256·35-s − 0.0401·37-s + 1.74·39-s + 1.10·41-s + 1.46·43-s + 0.323·45-s + 0.848·47-s − 0.616·49-s + 0.692·51-s + 0.295·53-s − 0.280·55-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 \( 1 + 2.31T + 3T^{2} \)
5 \( 1 - 0.927T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + 4.71T + 13T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 - 6.96T + 19T^{2} \)
23 \( 1 - 5.04T + 23T^{2} \)
29 \( 1 + 5.17T + 29T^{2} \)
31 \( 1 + 4.00T + 31T^{2} \)
37 \( 1 + 0.244T + 37T^{2} \)
41 \( 1 - 7.08T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 - 5.81T + 47T^{2} \)
53 \( 1 - 2.15T + 53T^{2} \)
59 \( 1 + 4.38T + 59T^{2} \)
61 \( 1 + 5.54T + 61T^{2} \)
67 \( 1 - 6.00T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 - 7.67T + 73T^{2} \)
79 \( 1 + 6.29T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−7.46647851277140367310333967051, −6.81659034969284761839663231152, −5.77412168735096105731138709321, −5.47790251693460396808898797945, −4.93195396925452443104257223903, −4.21409655693085240913428366383, −2.96317795520191482591856016738, −2.15282148247476354826662880692, −1.07220362467879091832477961900, 0, 1.07220362467879091832477961900, 2.15282148247476354826662880692, 2.96317795520191482591856016738, 4.21409655693085240913428366383, 4.93195396925452443104257223903, 5.47790251693460396808898797945, 5.77412168735096105731138709321, 6.81659034969284761839663231152, 7.46647851277140367310333967051

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