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 − 2·7-s − 8-s − 3·11-s − 2·13-s + 2·14-s + 16-s + 3·17-s + 7·19-s + 3·22-s + 4·23-s + 2·26-s − 2·28-s + 6·29-s + 4·31-s − 32-s − 3·34-s − 4·37-s − 7·38-s − 7·41-s − 4·43-s − 3·44-s − 4·46-s + 10·47-s − 3·49-s − 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.60·19-s + 0.639·22-s + 0.834·23-s + 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s − 1.13·38-s − 1.09·41-s − 0.609·43-s − 0.452·44-s − 0.589·46-s + 1.45·47-s − 3/7·49-s − 0.277·52-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  =  $(2,\ 75150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.513524501$
$L(\frac12)$  $\approx$  $1.513524501$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−13.91525119645892, −13.54866768291898, −13.18038975329739, −12.25624117745770, −12.11132340116979, −11.73002166442120, −10.80058635024664, −10.45387827170772, −10.02366491527374, −9.577825960842205, −9.072686046816612, −8.503476115711222, −7.794224443302800, −7.602152528036515, −6.761658231812063, −6.635381011575865, −5.614084064839775, −5.296479425599257, −4.753186109083122, −3.754834647951125, −3.073829398192578, −2.844499919129130, −2.021469180571614, −1.054334296229125, −0.5318563058479643, 0.5318563058479643, 1.054334296229125, 2.021469180571614, 2.844499919129130, 3.073829398192578, 3.754834647951125, 4.753186109083122, 5.296479425599257, 5.614084064839775, 6.635381011575865, 6.761658231812063, 7.602152528036515, 7.794224443302800, 8.503476115711222, 9.072686046816612, 9.577825960842205, 10.02366491527374, 10.45387827170772, 10.80058635024664, 11.73002166442120, 12.11132340116979, 12.25624117745770, 13.18038975329739, 13.54866768291898, 13.91525119645892

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