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  − 3.18·3-s − 1.57·5-s + 3.06·7-s + 7.15·9-s − 0.545·11-s − 1.64·13-s + 5.00·15-s − 3.65·17-s + 2.61·19-s − 9.75·21-s + 3.92·23-s − 2.53·25-s − 13.2·27-s + 7.11·29-s + 7.75·31-s + 1.73·33-s − 4.80·35-s − 9.57·37-s + 5.24·39-s − 1.04·41-s − 10.0·43-s − 11.2·45-s + 2.76·47-s + 2.36·49-s + 11.6·51-s − 9.38·53-s + 0.856·55-s + ⋯
L(s)  = 1  − 1.83·3-s − 0.702·5-s + 1.15·7-s + 2.38·9-s − 0.164·11-s − 0.456·13-s + 1.29·15-s − 0.885·17-s + 0.599·19-s − 2.12·21-s + 0.819·23-s − 0.506·25-s − 2.54·27-s + 1.32·29-s + 1.39·31-s + 0.302·33-s − 0.812·35-s − 1.57·37-s + 0.840·39-s − 0.162·41-s − 1.53·43-s − 1.67·45-s + 0.402·47-s + 0.338·49-s + 1.62·51-s − 1.28·53-s + 0.115·55-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 \( 1 + 3.18T + 3T^{2} \)
5 \( 1 + 1.57T + 5T^{2} \)
7 \( 1 - 3.06T + 7T^{2} \)
11 \( 1 + 0.545T + 11T^{2} \)
13 \( 1 + 1.64T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 - 7.11T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 9.57T + 37T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 2.76T + 47T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 - 2.80T + 59T^{2} \)
61 \( 1 - 5.26T + 61T^{2} \)
67 \( 1 + 7.28T + 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−7.24290499596383716486917583466, −6.83260716216865342930723496881, −6.07019013957762556878050412487, −5.21713545902187066949970689897, −4.73648427851435667506557674910, −4.41781820887944097940646749104, −3.23089224787951521218034080318, −1.89546937511164083850552340752, −1.00622329637241671208116431983, 0, 1.00622329637241671208116431983, 1.89546937511164083850552340752, 3.23089224787951521218034080318, 4.41781820887944097940646749104, 4.73648427851435667506557674910, 5.21713545902187066949970689897, 6.07019013957762556878050412487, 6.83260716216865342930723496881, 7.24290499596383716486917583466

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