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  − 1.88·2-s + 2.22·3-s + 1.54·4-s + 0.640·5-s − 4.19·6-s − 4.86·7-s + 0.850·8-s + 1.96·9-s − 1.20·10-s + 4.50·11-s + 3.45·12-s − 0.00706·13-s + 9.16·14-s + 1.42·15-s − 4.69·16-s − 6.43·17-s − 3.70·18-s − 19-s + 0.992·20-s − 10.8·21-s − 8.49·22-s − 0.234·23-s + 1.89·24-s − 4.58·25-s + 0.0133·26-s − 2.30·27-s − 7.53·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 1.28·3-s + 0.774·4-s + 0.286·5-s − 1.71·6-s − 1.83·7-s + 0.300·8-s + 0.655·9-s − 0.381·10-s + 1.35·11-s + 0.995·12-s − 0.00195·13-s + 2.44·14-s + 0.368·15-s − 1.17·16-s − 1.56·17-s − 0.872·18-s − 0.229·19-s + 0.221·20-s − 2.36·21-s − 1.81·22-s − 0.0488·23-s + 0.387·24-s − 0.917·25-s + 0.00260·26-s − 0.443·27-s − 1.42·28-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 \( 1 + 1.88T + 2T^{2} \)
3 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 - 0.640T + 5T^{2} \)
7 \( 1 + 4.86T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + 0.00706T + 13T^{2} \)
17 \( 1 + 6.43T + 17T^{2} \)
23 \( 1 + 0.234T + 23T^{2} \)
29 \( 1 - 8.13T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 1.87T + 37T^{2} \)
41 \( 1 - 0.709T + 41T^{2} \)
43 \( 1 - 2.98T + 43T^{2} \)
47 \( 1 + 8.81T + 47T^{2} \)
53 \( 1 - 6.85T + 53T^{2} \)
59 \( 1 + 9.51T + 59T^{2} \)
61 \( 1 - 2.64T + 61T^{2} \)
67 \( 1 - 5.68T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 3.68T + 73T^{2} \)
79 \( 1 + 9.08T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 9.09T + 89T^{2} \)
97 \( 1 + 7.06T + 97T^{2} \)
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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

−8.098452916113819359569383313217, −7.05135810050656989016470048110, −6.58671922613647329679862877197, −6.12590168655711934484766640267, −4.44662674157611688536597451235, −3.90706169784801961478864918825, −2.87445147896953875772996176796, −2.35966408425215181696714464887, −1.24445322398853853526178090195, 0, 1.24445322398853853526178090195, 2.35966408425215181696714464887, 2.87445147896953875772996176796, 3.90706169784801961478864918825, 4.44662674157611688536597451235, 6.12590168655711934484766640267, 6.58671922613647329679862877197, 7.05135810050656989016470048110, 8.098452916113819359569383313217

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