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 − 4·7-s − 8-s − 6·11-s − 13-s + 4·14-s + 16-s + 6·17-s − 19-s + 6·22-s + 6·23-s + 26-s − 4·28-s + 3·29-s + 2·31-s − 32-s − 6·34-s − 7·37-s + 38-s + 3·41-s − 10·43-s − 6·44-s − 6·46-s + 9·49-s − 52-s − 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.229·19-s + 1.27·22-s + 1.25·23-s + 0.196·26-s − 0.755·28-s + 0.557·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 1.15·37-s + 0.162·38-s + 0.468·41-s − 1.52·43-s − 0.904·44-s − 0.884·46-s + 9/7·49-s − 0.138·52-s − 1.64·53-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  =  $(2,\ 75150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5816224499$
$L(\frac12)$  $\approx$  $0.5816224499$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.99360533116797, −13.47586678865457, −12.83893205873932, −12.70101435625967, −12.17917200749595, −11.50463072010734, −10.84428278596337, −10.40645353081790, −9.948118554788860, −9.724248709398276, −9.080781321707166, −8.337566316907149, −8.079618937591215, −7.378763407756511, −6.924730456525246, −6.460703759501670, −5.736676931283679, −5.260562562284004, −4.772523760683561, −3.656518681890791, −3.109235163320495, −2.850525798249979, −2.084869687614273, −1.086631663142105, −0.3118576028522019, 0.3118576028522019, 1.086631663142105, 2.084869687614273, 2.850525798249979, 3.109235163320495, 3.656518681890791, 4.772523760683561, 5.260562562284004, 5.736676931283679, 6.460703759501670, 6.924730456525246, 7.378763407756511, 8.079618937591215, 8.337566316907149, 9.080781321707166, 9.724248709398276, 9.948118554788860, 10.40645353081790, 10.84428278596337, 11.50463072010734, 12.17917200749595, 12.70101435625967, 12.83893205873932, 13.47586678865457, 13.99360533116797

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