Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 223 $
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-s + 4-s − 2.38·5-s + 1.61·7-s + 8-s − 2.38·10-s − 1.29·11-s − 6.29·13-s + 1.61·14-s + 16-s + 6.73·17-s − 4.10·19-s − 2.38·20-s − 1.29·22-s + 8.82·23-s + 0.677·25-s − 6.29·26-s + 1.61·28-s − 3.74·29-s + 2.10·31-s + 32-s + 6.73·34-s − 3.84·35-s − 9.37·37-s − 4.10·38-s − 2.38·40-s − 8.42·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.06·5-s + 0.609·7-s + 0.353·8-s − 0.753·10-s − 0.390·11-s − 1.74·13-s + 0.431·14-s + 0.250·16-s + 1.63·17-s − 0.941·19-s − 0.532·20-s − 0.275·22-s + 1.84·23-s + 0.135·25-s − 1.23·26-s + 0.304·28-s − 0.694·29-s + 0.377·31-s + 0.176·32-s + 1.15·34-s − 0.649·35-s − 1.54·37-s − 0.665·38-s − 0.376·40-s − 1.31·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4014} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4014,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;223\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;223\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 + 2.38T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 1.29T + 11T^{2} \)
13 \( 1 + 6.29T + 13T^{2} \)
17 \( 1 - 6.73T + 17T^{2} \)
19 \( 1 + 4.10T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 - 2.10T + 31T^{2} \)
37 \( 1 + 9.37T + 37T^{2} \)
41 \( 1 + 8.42T + 41T^{2} \)
43 \( 1 + 3.32T + 43T^{2} \)
47 \( 1 - 1.48T + 47T^{2} \)
53 \( 1 + 7.88T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 4.37T + 61T^{2} \)
67 \( 1 + 3.61T + 67T^{2} \)
71 \( 1 + 2.01T + 71T^{2} \)
73 \( 1 - 7.03T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 4.90T + 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.008963575761621289097373140156, −7.20035685190131497420655058385, −6.90599939516568157039641722759, −5.45004453513696317635018713083, −5.09953041377310457369382716618, −4.35672888197568198243809482881, −3.43642762080122250982860197875, −2.74070500089549069615216174380, −1.56051860456743620234937186886, 0, 1.56051860456743620234937186886, 2.74070500089549069615216174380, 3.43642762080122250982860197875, 4.35672888197568198243809482881, 5.09953041377310457369382716618, 5.45004453513696317635018713083, 6.90599939516568157039641722759, 7.20035685190131497420655058385, 8.008963575761621289097373140156

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