Properties

Degree 2
Conductor $ 19 \cdot 317 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.20·2-s − 1.66·3-s + 2.86·4-s + 2.85·5-s + 3.67·6-s − 0.996·7-s − 1.91·8-s − 0.226·9-s − 6.30·10-s + 4.63·11-s − 4.77·12-s + 1.51·13-s + 2.19·14-s − 4.76·15-s − 1.50·16-s + 2.86·17-s + 0.499·18-s − 19-s + 8.20·20-s + 1.65·21-s − 10.2·22-s − 4.55·23-s + 3.19·24-s + 3.17·25-s − 3.33·26-s + 5.37·27-s − 2.85·28-s + ⋯
L(s)  = 1  − 1.56·2-s − 0.961·3-s + 1.43·4-s + 1.27·5-s + 1.50·6-s − 0.376·7-s − 0.677·8-s − 0.0754·9-s − 1.99·10-s + 1.39·11-s − 1.37·12-s + 0.418·13-s + 0.587·14-s − 1.22·15-s − 0.377·16-s + 0.693·17-s + 0.117·18-s − 0.229·19-s + 1.83·20-s + 0.362·21-s − 2.18·22-s − 0.948·23-s + 0.651·24-s + 0.635·25-s − 0.653·26-s + 1.03·27-s − 0.540·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6023\)    =    \(19 \cdot 317\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6023} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6023,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;317\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;317\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 + T \)
317 \( 1 + T \)
good2 \( 1 + 2.20T + 2T^{2} \)
3 \( 1 + 1.66T + 3T^{2} \)
5 \( 1 - 2.85T + 5T^{2} \)
7 \( 1 + 0.996T + 7T^{2} \)
11 \( 1 - 4.63T + 11T^{2} \)
13 \( 1 - 1.51T + 13T^{2} \)
17 \( 1 - 2.86T + 17T^{2} \)
23 \( 1 + 4.55T + 23T^{2} \)
29 \( 1 + 8.47T + 29T^{2} \)
31 \( 1 - 9.31T + 31T^{2} \)
37 \( 1 - 3.67T + 37T^{2} \)
41 \( 1 - 4.17T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 7.87T + 53T^{2} \)
59 \( 1 - 4.45T + 59T^{2} \)
61 \( 1 - 0.678T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 1.30T + 71T^{2} \)
73 \( 1 + 5.52T + 73T^{2} \)
79 \( 1 + 9.51T + 79T^{2} \)
83 \( 1 + 4.61T + 83T^{2} \)
89 \( 1 + 6.53T + 89T^{2} \)
97 \( 1 + 3.54T + 97T^{2} \)
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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

−7.959146538606600131416541369280, −6.81225182850096456355916149273, −6.39224842895976764699889431390, −5.99029784792353893035237397865, −5.14111455126807923131842327649, −4.05310222582801917438603417046, −2.87839457399224416536118988982, −1.74390221577240608516529899952, −1.21629356831011113375148286074, 0, 1.21629356831011113375148286074, 1.74390221577240608516529899952, 2.87839457399224416536118988982, 4.05310222582801917438603417046, 5.14111455126807923131842327649, 5.99029784792353893035237397865, 6.39224842895976764699889431390, 6.81225182850096456355916149273, 7.959146538606600131416541369280

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