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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 2·11-s + 6·13-s − 14-s + 16-s + 6·17-s + 19-s + 2·22-s + 3·23-s − 6·26-s + 28-s + 2·29-s − 10·31-s − 32-s − 6·34-s + 2·37-s − 38-s − 9·41-s + 8·43-s − 2·44-s − 3·46-s + 8·47-s − 6·49-s + 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.426·22-s + 0.625·23-s − 1.17·26-s + 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.162·38-s − 1.40·41-s + 1.21·43-s − 0.301·44-s − 0.442·46-s + 1.16·47-s − 6/7·49-s + 0.832·52-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  =  0
Selberg data  =  $(2,\ 75150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.277665699$
$L(\frac12)$  $\approx$  $2.277665699$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.15596823036006, −13.46763926859860, −13.08087878475913, −12.60710993683609, −11.77958633219525, −11.63427691318255, −10.91619694863408, −10.48595475893873, −10.21225173381807, −9.439499214431272, −8.874039473266259, −8.567546781770708, −7.992577420799803, −7.380983822844583, −7.144914371019751, −6.250134882570140, −5.660998476434981, −5.457175220632354, −4.589669002470346, −3.742868711633417, −3.374845947770883, −2.662892320318091, −1.824061341622156, −1.218101253019796, −0.6207880918755594, 0.6207880918755594, 1.218101253019796, 1.824061341622156, 2.662892320318091, 3.374845947770883, 3.742868711633417, 4.589669002470346, 5.457175220632354, 5.660998476434981, 6.250134882570140, 7.144914371019751, 7.380983822844583, 7.992577420799803, 8.567546781770708, 8.874039473266259, 9.439499214431272, 10.21225173381807, 10.48595475893873, 10.91619694863408, 11.63427691318255, 11.77958633219525, 12.60710993683609, 13.08087878475913, 13.46763926859860, 14.15596823036006

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