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 − 3·7-s − 8-s − 6·11-s − 2·13-s + 3·14-s + 16-s + 6·22-s + 4·23-s + 2·26-s − 3·28-s + 6·29-s − 9·31-s − 32-s + 37-s − 2·41-s + 8·43-s − 6·44-s − 4·46-s − 47-s + 2·49-s − 2·52-s − 3·53-s + 3·56-s − 6·58-s − 3·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 1.80·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s + 1.27·22-s + 0.834·23-s + 0.392·26-s − 0.566·28-s + 1.11·29-s − 1.61·31-s − 0.176·32-s + 0.164·37-s − 0.312·41-s + 1.21·43-s − 0.904·44-s − 0.589·46-s − 0.145·47-s + 2/7·49-s − 0.277·52-s − 0.412·53-s + 0.400·56-s − 0.787·58-s − 0.390·59-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
167 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.45661705858566, −13.65688494680118, −13.15490623613226, −12.87544475488240, −12.34293338812705, −11.89154708277119, −11.07467049078352, −10.61494179867289, −10.36804723053120, −9.785005703559868, −9.214661938533717, −8.916512071732878, −8.155392985022382, −7.553448131869346, −7.365030949692368, −6.645408462634632, −6.083913335933820, −5.531807466683750, −4.953832527560460, −4.337246627844637, −3.288032935332837, −2.995037299783418, −2.447289485191761, −1.675731449587168, −0.6123941925813091, 0, 0.6123941925813091, 1.675731449587168, 2.447289485191761, 2.995037299783418, 3.288032935332837, 4.337246627844637, 4.953832527560460, 5.531807466683750, 6.083913335933820, 6.645408462634632, 7.365030949692368, 7.553448131869346, 8.155392985022382, 8.916512071732878, 9.214661938533717, 9.785005703559868, 10.36804723053120, 10.61494179867289, 11.07467049078352, 11.89154708277119, 12.34293338812705, 12.87544475488240, 13.15490623613226, 13.65688494680118, 14.45661705858566

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