Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
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 − 7-s + 8-s + 2·11-s − 7·13-s − 14-s + 16-s − 7·17-s + 8·19-s + 2·22-s − 5·23-s − 7·26-s − 28-s − 9·29-s + 31-s + 32-s − 7·34-s + 2·37-s + 8·38-s − 11·41-s − 3·43-s + 2·44-s − 5·46-s + 4·47-s + 49-s − 7·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.603·11-s − 1.94·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s + 1.83·19-s + 0.426·22-s − 1.04·23-s − 1.37·26-s − 0.188·28-s − 1.67·29-s + 0.179·31-s + 0.176·32-s − 1.20·34-s + 0.328·37-s + 1.29·38-s − 1.71·41-s − 0.457·43-s + 0.301·44-s − 0.737·46-s + 0.583·47-s + 1/7·49-s − 0.970·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−18.88051878351064, −18.27494064441225, −17.41452755761013, −16.98526579981058, −16.28187757617455, −15.57367380255398, −15.07098957072685, −14.40734302157928, −13.75116872813311, −13.24514428761039, −12.42688427877754, −11.84131175103283, −11.44368155026687, −10.45086959498909, −9.632578686683150, −9.319154229142094, −8.166195220376784, −7.225318687090355, −6.948488908149546, −5.928884257733280, −5.171644336234710, −4.451413238594026, −3.597056748937578, −2.651590475962542, −1.794042241139142, 0, 1.794042241139142, 2.651590475962542, 3.597056748937578, 4.451413238594026, 5.171644336234710, 5.928884257733280, 6.948488908149546, 7.225318687090355, 8.166195220376784, 9.319154229142094, 9.632578686683150, 10.45086959498909, 11.44368155026687, 11.84131175103283, 12.42688427877754, 13.24514428761039, 13.75116872813311, 14.40734302157928, 15.07098957072685, 15.57367380255398, 16.28187757617455, 16.98526579981058, 17.41452755761013, 18.27494064441225, 18.88051878351064

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