Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 167 $
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 + 4·7-s − 8-s + 4·11-s − 4·14-s + 16-s − 4·17-s − 4·19-s − 4·22-s − 4·23-s + 4·28-s + 2·29-s + 4·31-s − 32-s + 4·34-s + 12·37-s + 4·38-s − 12·41-s + 8·43-s + 4·44-s + 4·46-s + 9·49-s + 14·53-s − 4·56-s − 2·58-s − 2·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 1.20·11-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.917·19-s − 0.852·22-s − 0.834·23-s + 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s + 1.97·37-s + 0.648·38-s − 1.87·41-s + 1.21·43-s + 0.603·44-s + 0.589·46-s + 9/7·49-s + 1.92·53-s − 0.534·56-s − 0.262·58-s − 0.260·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(75150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{75150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 75150,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
167 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−14.50790440594903, −13.82169698300480, −13.46082315885045, −12.73022248051213, −12.03359058486390, −11.69149485862450, −11.38786452232722, −10.78473249639909, −10.34314887854910, −9.784091832292535, −9.101567120068632, −8.647971776877771, −8.374792277756544, −7.742471295051042, −7.268625684037412, −6.482809467752777, −6.287592775398855, −5.490919962996515, −4.780022756757390, −4.140051155900834, −3.977728187068545, −2.685844546353415, −2.281338278246583, −1.505704890407836, −1.085608700864595, 0, 1.085608700864595, 1.505704890407836, 2.281338278246583, 2.685844546353415, 3.977728187068545, 4.140051155900834, 4.780022756757390, 5.490919962996515, 6.287592775398855, 6.482809467752777, 7.268625684037412, 7.742471295051042, 8.374792277756544, 8.647971776877771, 9.101567120068632, 9.784091832292535, 10.34314887854910, 10.78473249639909, 11.38786452232722, 11.69149485862450, 12.03359058486390, 12.73022248051213, 13.46082315885045, 13.82169698300480, 14.50790440594903

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