Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17 $
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 + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s + 17-s − 4·19-s + 20-s + 25-s − 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s − 34-s + 35-s + 2·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s − 12·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10710} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 10710,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.97931368584263, −16.41820876856211, −15.70516369782868, −15.19086645928072, −14.62446322968118, −14.04958301496418, −13.26712575891940, −12.93445912519119, −12.03391476296415, −11.54421269729425, −10.91136744072833, −10.38792362437440, −9.778887791804477, −9.208857765924295, −8.479895938198790, −8.088111019991270, −7.386757828468904, −6.426465118361218, −6.253528560414446, −5.257563740393668, −4.602785458451671, −3.642942152715429, −2.846090608724181, −1.905005984060601, −1.297361900472208, 0, 1.297361900472208, 1.905005984060601, 2.846090608724181, 3.642942152715429, 4.602785458451671, 5.257563740393668, 6.253528560414446, 6.426465118361218, 7.386757828468904, 8.088111019991270, 8.479895938198790, 9.208857765924295, 9.778887791804477, 10.38792362437440, 10.91136744072833, 11.54421269729425, 12.03391476296415, 12.93445912519119, 13.26712575891940, 14.04958301496418, 14.62446322968118, 15.19086645928072, 15.70516369782868, 16.41820876856211, 16.97931368584263

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