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 − 8-s + 5·11-s + 5·13-s + 16-s + 5·17-s + 5·19-s − 5·22-s + 3·23-s − 5·26-s + 29-s − 7·31-s − 32-s − 5·34-s + 11·37-s − 5·38-s − 6·41-s + 8·43-s + 5·44-s − 3·46-s + 2·47-s − 7·49-s + 5·52-s − 10·53-s − 58-s + 6·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.50·11-s + 1.38·13-s + 1/4·16-s + 1.21·17-s + 1.14·19-s − 1.06·22-s + 0.625·23-s − 0.980·26-s + 0.185·29-s − 1.25·31-s − 0.176·32-s − 0.857·34-s + 1.80·37-s − 0.811·38-s − 0.937·41-s + 1.21·43-s + 0.753·44-s − 0.442·46-s + 0.291·47-s − 49-s + 0.693·52-s − 1.37·53-s − 0.131·58-s + 0.781·59-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  =  0
Selberg data  =  $(2,\ 75150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.798437239$
$L(\frac12)$  $\approx$  $2.798437239$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−14.15108686575358, −13.64781553411536, −13.00201991469186, −12.48134365752327, −11.95254074999013, −11.42412660571086, −11.11773075530750, −10.62927717797028, −9.816118791725664, −9.445063244115592, −9.150365438495437, −8.519004349417909, −7.908780919887990, −7.567356721358049, −6.754476943831510, −6.497698515042845, −5.716598745182807, −5.453359799156468, −4.438487842582139, −3.833203395631890, −3.318584992869943, −2.798039019832369, −1.610293498934585, −1.306489630724821, −0.7008471082744707, 0.7008471082744707, 1.306489630724821, 1.610293498934585, 2.798039019832369, 3.318584992869943, 3.833203395631890, 4.438487842582139, 5.453359799156468, 5.716598745182807, 6.497698515042845, 6.754476943831510, 7.567356721358049, 7.908780919887990, 8.519004349417909, 9.150365438495437, 9.445063244115592, 9.816118791725664, 10.62927717797028, 11.11773075530750, 11.42412660571086, 11.95254074999013, 12.48134365752327, 13.00201991469186, 13.64781553411536, 14.15108686575358

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