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 − 3·7-s − 8-s − 2·11-s + 6·13-s + 3·14-s + 16-s − 2·17-s + 19-s + 2·22-s − 3·23-s − 6·26-s − 3·28-s − 6·29-s + 2·31-s − 32-s + 2·34-s + 2·37-s − 38-s + 5·41-s + 8·43-s − 2·44-s + 3·46-s − 8·47-s + 2·49-s + 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.801·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s + 0.426·22-s − 0.625·23-s − 1.17·26-s − 0.566·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.162·38-s + 0.780·41-s + 1.21·43-s − 0.301·44-s + 0.442·46-s − 1.16·47-s + 2/7·49-s + 0.832·52-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−14.15813568422644, −13.83029351572967, −13.19047783319527, −12.77591324770374, −12.52883275026744, −11.53384735490169, −11.28672996301354, −10.76737876774603, −10.26828458725348, −9.699420549364091, −9.267953738709308, −8.853593705684484, −8.168315713042959, −7.793239882123756, −7.190375806556391, −6.423222421574095, −6.209449615628719, −5.711833286304847, −4.928029237997891, −4.053881443787461, −3.589324142754703, −3.004632121254767, −2.334604602470011, −1.590430932490741, −0.7734915648694529, 0, 0.7734915648694529, 1.590430932490741, 2.334604602470011, 3.004632121254767, 3.589324142754703, 4.053881443787461, 4.928029237997891, 5.711833286304847, 6.209449615628719, 6.423222421574095, 7.190375806556391, 7.793239882123756, 8.168315713042959, 8.853593705684484, 9.267953738709308, 9.699420549364091, 10.26828458725348, 10.76737876774603, 11.28672996301354, 11.53384735490169, 12.52883275026744, 12.77591324770374, 13.19047783319527, 13.83029351572967, 14.15813568422644

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