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 − 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s − 2·26-s − 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s − 2·37-s − 4·38-s − 6·41-s − 8·43-s − 12·47-s + 49-s − 2·52-s + 6·53-s − 56-s + 6·58-s + 12·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s − 0.133·56-s + 0.787·58-s + 1.56·59-s + 0.256·61-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−8.441852246728577993237698406657, −7.27859350746563522747987197323, −6.71638041353320182294207342507, −6.11443767828676558964940622186, −5.06486144035934139721922770973, −4.49511516695984015657784194328, −3.59130715422509910600760079256, −2.65322176991464818400250263188, −1.79300221067561004266493520841, 0, 1.79300221067561004266493520841, 2.65322176991464818400250263188, 3.59130715422509910600760079256, 4.49511516695984015657784194328, 5.06486144035934139721922770973, 6.11443767828676558964940622186, 6.71638041353320182294207342507, 7.27859350746563522747987197323, 8.441852246728577993237698406657

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