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  + 0.791·3-s + 0.178·5-s + 0.0809·7-s − 2.37·9-s + 4.30·11-s + 1.10·13-s + 0.141·15-s − 3.06·17-s + 2.15·19-s + 0.0640·21-s − 9.15·23-s − 4.96·25-s − 4.25·27-s − 6.17·29-s + 9.82·31-s + 3.40·33-s + 0.0144·35-s + 4.08·37-s + 0.871·39-s + 4.58·41-s − 2.03·43-s − 0.424·45-s + 5.12·47-s − 6.99·49-s − 2.42·51-s + 2.69·53-s + 0.769·55-s + ⋯
L(s)  = 1  + 0.457·3-s + 0.0799·5-s + 0.0305·7-s − 0.791·9-s + 1.29·11-s + 0.305·13-s + 0.0365·15-s − 0.743·17-s + 0.495·19-s + 0.0139·21-s − 1.90·23-s − 0.993·25-s − 0.818·27-s − 1.14·29-s + 1.76·31-s + 0.593·33-s + 0.00244·35-s + 0.671·37-s + 0.139·39-s + 0.716·41-s − 0.309·43-s − 0.0632·45-s + 0.748·47-s − 0.999·49-s − 0.339·51-s + 0.369·53-s + 0.103·55-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 \( 1 - 0.791T + 3T^{2} \)
5 \( 1 - 0.178T + 5T^{2} \)
7 \( 1 - 0.0809T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
13 \( 1 - 1.10T + 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 - 2.15T + 19T^{2} \)
23 \( 1 + 9.15T + 23T^{2} \)
29 \( 1 + 6.17T + 29T^{2} \)
31 \( 1 - 9.82T + 31T^{2} \)
37 \( 1 - 4.08T + 37T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 + 2.03T + 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 - 2.69T + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 + 5.62T + 67T^{2} \)
71 \( 1 - 4.88T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 - 5.57T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−7.78000963423418708688024436398, −6.64743302440547111249649712811, −6.11839407206300306288846523488, −5.61423608221065638469053335887, −4.37562784754540117901587333437, −3.97813460461231242897749855162, −3.09704685421057847738343832529, −2.23379838047914627923596684579, −1.41187947184039973695053820193, 0, 1.41187947184039973695053820193, 2.23379838047914627923596684579, 3.09704685421057847738343832529, 3.97813460461231242897749855162, 4.37562784754540117901587333437, 5.61423608221065638469053335887, 6.11839407206300306288846523488, 6.64743302440547111249649712811, 7.78000963423418708688024436398

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