Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13 $
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 − 8-s − 4·11-s − 13-s + 16-s − 4·17-s + 2·19-s + 4·22-s + 2·23-s + 26-s − 8·29-s − 4·31-s − 32-s + 4·34-s − 6·37-s − 2·38-s + 10·41-s − 4·43-s − 4·44-s − 2·46-s − 52-s + 6·53-s + 8·58-s − 12·59-s + 2·61-s + 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 0.970·17-s + 0.458·19-s + 0.852·22-s + 0.417·23-s + 0.196·26-s − 1.48·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.986·37-s − 0.324·38-s + 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.294·46-s − 0.138·52-s + 0.824·53-s + 1.05·58-s − 1.56·59-s + 0.256·61-s + 0.508·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(286650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{286650} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 286650,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−12.96040930421340, −12.56460404739960, −11.93817794949377, −11.47110051164843, −10.92634414408111, −10.76705585433149, −10.23314927234193, −9.653978194409140, −9.251436046754857, −8.902317155594565, −8.299404138859516, −7.863964745973679, −7.280958995544387, −7.177212602591428, −6.453724340448112, −5.849517595616158, −5.405622867774526, −4.974848464215577, −4.332751351985412, −3.673353005635891, −3.131062720996985, −2.500861870139196, −2.079011844506654, −1.478111838170312, −0.5757062793651180, 0, 0.5757062793651180, 1.478111838170312, 2.079011844506654, 2.500861870139196, 3.131062720996985, 3.673353005635891, 4.332751351985412, 4.974848464215577, 5.405622867774526, 5.849517595616158, 6.453724340448112, 7.177212602591428, 7.280958995544387, 7.863964745973679, 8.299404138859516, 8.902317155594565, 9.251436046754857, 9.653978194409140, 10.23314927234193, 10.76705585433149, 10.92634414408111, 11.47110051164843, 11.93817794949377, 12.56460404739960, 12.96040930421340

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