Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 13^{2} $
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 − 3-s − 4-s − 5-s − 6-s − 3·8-s + 9-s − 10-s + 4·11-s + 12-s + 15-s − 16-s + 2·17-s + 18-s − 4·19-s + 20-s + 4·22-s + 3·24-s + 25-s − 27-s − 2·29-s + 30-s + 5·32-s − 4·33-s + 2·34-s − 36-s + 10·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.182·30-s + 0.883·32-s − 0.696·33-s + 0.342·34-s − 1/6·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2535} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2535,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.14291087682512, −18.70042502794435, −17.91647706226479, −17.32128966965020, −16.73889600619369, −16.12382950179161, −15.17904108844575, −14.70770442171207, −14.23792455632818, −13.26716053005862, −12.83231993162115, −12.07387517580031, −11.62482837712158, −10.92758744187718, −9.910992058599380, −9.333388465635555, −8.526905706632633, −7.769107503680429, −6.630784393805768, −6.204816492497732, −5.271520767972770, −4.463429909156866, −3.922652871976597, −3.017469807948261, −1.418360196760929, 0, 1.418360196760929, 3.017469807948261, 3.922652871976597, 4.463429909156866, 5.271520767972770, 6.204816492497732, 6.630784393805768, 7.769107503680429, 8.526905706632633, 9.333388465635555, 9.910992058599380, 10.92758744187718, 11.62482837712158, 12.07387517580031, 12.83231993162115, 13.26716053005862, 14.23792455632818, 14.70770442171207, 15.17904108844575, 16.12382950179161, 16.73889600619369, 17.32128966965020, 17.91647706226479, 18.70042502794435, 19.14291087682512

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