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 + 2·7-s − 8-s + 3·11-s + 3·13-s − 2·14-s + 16-s + 3·17-s − 19-s − 3·22-s − 9·23-s − 3·26-s + 2·28-s + 5·29-s − 31-s − 32-s − 3·34-s + 5·37-s + 38-s − 12·41-s + 4·43-s + 3·44-s + 9·46-s − 8·47-s − 3·49-s + 3·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s + 0.832·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.639·22-s − 1.87·23-s − 0.588·26-s + 0.377·28-s + 0.928·29-s − 0.179·31-s − 0.176·32-s − 0.514·34-s + 0.821·37-s + 0.162·38-s − 1.87·41-s + 0.609·43-s + 0.452·44-s + 1.32·46-s − 1.16·47-s − 3/7·49-s + 0.416·52-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  =  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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.27252594753693, −14.04778914335298, −13.36775239765187, −12.66406723044353, −12.20529094365703, −11.61653149959905, −11.40532135437667, −10.82229697938324, −10.16792344724886, −9.784852355426117, −9.359990028074672, −8.457589598600491, −8.287041582284197, −7.967057939703687, −7.142515599098163, −6.528133300487437, −6.228367540494110, −5.511157088258270, −4.932833972029435, −4.090251186046917, −3.752210101052251, −2.977528887820883, −2.126108904794057, −1.535330493039106, −1.052019749338507, 0, 1.052019749338507, 1.535330493039106, 2.126108904794057, 2.977528887820883, 3.752210101052251, 4.090251186046917, 4.932833972029435, 5.511157088258270, 6.228367540494110, 6.528133300487437, 7.142515599098163, 7.967057939703687, 8.287041582284197, 8.457589598600491, 9.359990028074672, 9.784852355426117, 10.16792344724886, 10.82229697938324, 11.40532135437667, 11.61653149959905, 12.20529094365703, 12.66406723044353, 13.36775239765187, 14.04778914335298, 14.27252594753693

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