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 + 4·13-s − 4·14-s + 16-s + 4·19-s + 4·22-s + 4·23-s − 4·26-s + 4·28-s + 2·29-s − 4·31-s − 32-s + 8·37-s − 4·38-s − 8·41-s − 8·43-s − 4·44-s − 4·46-s − 8·47-s + 9·49-s + 4·52-s − 2·53-s − 4·56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.20·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s + 0.917·19-s + 0.852·22-s + 0.834·23-s − 0.784·26-s + 0.755·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 1.31·37-s − 0.648·38-s − 1.24·41-s − 1.21·43-s − 0.603·44-s − 0.589·46-s − 1.16·47-s + 9/7·49-s + 0.554·52-s − 0.274·53-s − 0.534·56-s + ⋯

Functional equation

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

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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.51698192461308, −13.72189355007936, −13.29554450913568, −12.98514970675436, −12.07848038499488, −11.66469414306792, −11.14157537464262, −10.95008114143272, −10.30507988921258, −9.848865093266387, −9.182446287193948, −8.566941680040937, −8.187147492796998, −7.899664905047044, −7.241017826583611, −6.766404291399753, −5.965350886219347, −5.379807274018103, −5.004381074712636, −4.397812186096918, −3.485895097796214, −2.971183875967944, −2.210960860824540, −1.481222394417088, −1.080007653601440, 0, 1.080007653601440, 1.481222394417088, 2.210960860824540, 2.971183875967944, 3.485895097796214, 4.397812186096918, 5.004381074712636, 5.379807274018103, 5.965350886219347, 6.766404291399753, 7.241017826583611, 7.899664905047044, 8.187147492796998, 8.566941680040937, 9.182446287193948, 9.848865093266387, 10.30507988921258, 10.95008114143272, 11.14157537464262, 11.66469414306792, 12.07848038499488, 12.98514970675436, 13.29554450913568, 13.72189355007936, 14.51698192461308

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