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 + 4·13-s − 14-s + 16-s + 2·17-s − 8·19-s − 8·23-s − 4·26-s + 28-s − 8·29-s + 4·31-s − 32-s − 2·34-s − 8·37-s + 8·38-s − 12·41-s + 8·43-s + 8·46-s + 4·47-s + 49-s + 4·52-s + 6·53-s − 56-s + 8·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 1.66·23-s − 0.784·26-s + 0.188·28-s − 1.48·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s − 1.31·37-s + 1.29·38-s − 1.87·41-s + 1.21·43-s + 1.17·46-s + 0.583·47-s + 1/7·49-s + 0.554·52-s + 0.824·53-s − 0.133·56-s + 1.05·58-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  :  $\chi_{3150} (1, \cdot )$
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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.343257819346446236965751637523, −7.81206844218511378024997408335, −6.83373060181886013429823678971, −6.14420129225749751260411434902, −5.46283835623425377901995720059, −4.22820327125608953941410660969, −3.57966154683185305802343791768, −2.25154528347514162235849961741, −1.50341634374332771695850686693, 0, 1.50341634374332771695850686693, 2.25154528347514162235849961741, 3.57966154683185305802343791768, 4.22820327125608953941410660969, 5.46283835623425377901995720059, 6.14420129225749751260411434902, 6.83373060181886013429823678971, 7.81206844218511378024997408335, 8.343257819346446236965751637523

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