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  − 2.23·3-s + 1.23·5-s + 4.23·7-s + 2.00·9-s + 4.23·11-s − 5.47·13-s − 2.76·15-s + 2·17-s + 7.70·19-s − 9.47·21-s − 2.23·23-s − 3.47·25-s + 2.23·27-s − 2.76·29-s − 7.70·31-s − 9.47·33-s + 5.23·35-s − 4.47·37-s + 12.2·39-s − 5.23·41-s − 10.2·43-s + 2.47·45-s − 4.23·47-s + 10.9·49-s − 4.47·51-s − 0.472·53-s + 5.23·55-s + ⋯
L(s)  = 1  − 1.29·3-s + 0.552·5-s + 1.60·7-s + 0.666·9-s + 1.27·11-s − 1.51·13-s − 0.713·15-s + 0.485·17-s + 1.76·19-s − 2.06·21-s − 0.466·23-s − 0.694·25-s + 0.430·27-s − 0.513·29-s − 1.38·31-s − 1.64·33-s + 0.885·35-s − 0.735·37-s + 1.95·39-s − 0.817·41-s − 1.56·43-s + 0.368·45-s − 0.617·47-s + 1.56·49-s − 0.626·51-s − 0.0648·53-s + 0.706·55-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 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 - 4.23T + 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 + 5.47T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 7.70T + 19T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 7.47T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 + 0.763T + 71T^{2} \)
73 \( 1 + 7.52T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.25445600366208856916394748756, −6.91832467020040870539121620436, −5.75961316903722887355739237620, −5.46276534061150849656010852242, −4.92347604667195376247806181486, −4.19767104223064373053467955791, −3.12137151109480175688563203292, −1.72915620013153467827217334746, −1.44469799375910412984329104637, 0, 1.44469799375910412984329104637, 1.72915620013153467827217334746, 3.12137151109480175688563203292, 4.19767104223064373053467955791, 4.92347604667195376247806181486, 5.46276534061150849656010852242, 5.75961316903722887355739237620, 6.91832467020040870539121620436, 7.25445600366208856916394748756

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