Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23 $
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 + 7-s − 8-s + 9-s − 10-s + 12-s + 13-s − 14-s + 15-s + 16-s + 6·17-s − 18-s − 4·19-s + 20-s + 21-s − 23-s − 24-s + 25-s − 26-s + 27-s + 28-s − 6·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 62790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 62790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62790\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{62790} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 62790,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.58523640115189, −14.03899190503979, −13.59653642836513, −13.04714054159689, −12.37543973340944, −12.07901067079472, −11.36926154984007, −10.81434909377184, −10.35093353320404, −9.856402498268138, −9.427218827841159, −8.845880481343241, −8.305894753225063, −7.944912124657028, −7.391594632736021, −6.805120123957244, −6.124553697815872, −5.685816164545144, −5.015495517819509, −4.194349465323421, −3.702434308841582, −2.810821400679509, −2.463115732764304, −1.500590585183915, −1.203266409336467, 0, 1.203266409336467, 1.500590585183915, 2.463115732764304, 2.810821400679509, 3.702434308841582, 4.194349465323421, 5.015495517819509, 5.685816164545144, 6.124553697815872, 6.805120123957244, 7.391594632736021, 7.944912124657028, 8.305894753225063, 8.845880481343241, 9.427218827841159, 9.856402498268138, 10.35093353320404, 10.81434909377184, 11.36926154984007, 12.07901067079472, 12.37543973340944, 13.04714054159689, 13.59653642836513, 14.03899190503979, 14.58523640115189

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