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 + 8-s − 2·11-s + 6·13-s + 16-s − 4·17-s − 8·19-s − 2·22-s + 6·26-s + 6·29-s + 8·31-s + 32-s − 4·34-s − 2·37-s − 8·38-s − 10·41-s + 4·43-s − 2·44-s + 8·47-s − 7·49-s + 6·52-s − 4·53-s + 6·58-s + 12·59-s − 10·61-s + 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 1.66·13-s + 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.426·22-s + 1.17·26-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.685·34-s − 0.328·37-s − 1.29·38-s − 1.56·41-s + 0.609·43-s − 0.301·44-s + 1.16·47-s − 49-s + 0.832·52-s − 0.549·53-s + 0.787·58-s + 1.56·59-s − 1.28·61-s + 1.01·62-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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.18444806783537, −13.69131524775269, −13.26533240403062, −13.08531173443438, −12.30382045436337, −11.96669876319353, −11.24546755739086, −10.87028119690757, −10.42565342164547, −10.07381093600222, −9.046942121611005, −8.646051300048414, −8.271505061715295, −7.724196554782251, −6.801902267039344, −6.430531214969858, −6.187452366653392, −5.416364646250109, −4.724409098308974, −4.326422315248516, −3.773191917785765, −3.055384001197538, −2.466162388029965, −1.822865211595160, −1.035370397215068, 0, 1.035370397215068, 1.822865211595160, 2.466162388029965, 3.055384001197538, 3.773191917785765, 4.326422315248516, 4.724409098308974, 5.416364646250109, 6.187452366653392, 6.430531214969858, 6.801902267039344, 7.724196554782251, 8.271505061715295, 8.646051300048414, 9.046942121611005, 10.07381093600222, 10.42565342164547, 10.87028119690757, 11.24546755739086, 11.96669876319353, 12.30382045436337, 13.08531173443438, 13.26533240403062, 13.69131524775269, 14.18444806783537

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