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 − 5·11-s + 3·13-s − 4·14-s + 16-s + 7·17-s − 3·19-s − 5·22-s + 7·23-s + 3·26-s − 4·28-s + 9·29-s + 7·31-s + 32-s + 7·34-s + 5·37-s − 3·38-s − 8·41-s + 8·43-s − 5·44-s + 7·46-s + 8·47-s + 9·49-s + 3·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1.50·11-s + 0.832·13-s − 1.06·14-s + 1/4·16-s + 1.69·17-s − 0.688·19-s − 1.06·22-s + 1.45·23-s + 0.588·26-s − 0.755·28-s + 1.67·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s + 0.821·37-s − 0.486·38-s − 1.24·41-s + 1.21·43-s − 0.753·44-s + 1.03·46-s + 1.16·47-s + 9/7·49-s + 0.416·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

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  =  $(2,\ 75150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.731500041$
$L(\frac12)$  $\approx$  $3.731500041$
$L(\frac{3}{2})$   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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.91794891121409, −13.47386806239102, −13.08441909017114, −12.63680784061232, −12.30412281310285, −11.73014607173765, −10.99872862962156, −10.49065328894574, −10.10436167326140, −9.823494584353057, −8.849777870993120, −8.537244572770848, −7.783133730714325, −7.347221373321476, −6.694396486511627, −6.169606325759488, −5.819706170811305, −5.150151530048970, −4.649915476397325, −3.854163699481283, −3.285712517614169, −2.780342708215634, −2.497436901243318, −1.156237953722435, −0.6313640729599269, 0.6313640729599269, 1.156237953722435, 2.497436901243318, 2.780342708215634, 3.285712517614169, 3.854163699481283, 4.649915476397325, 5.150151530048970, 5.819706170811305, 6.169606325759488, 6.694396486511627, 7.347221373321476, 7.783133730714325, 8.537244572770848, 8.849777870993120, 9.823494584353057, 10.10436167326140, 10.49065328894574, 10.99872862962156, 11.73014607173765, 12.30412281310285, 12.63680784061232, 13.08441909017114, 13.47386806239102, 13.91794891121409

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