Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
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 − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s + 13-s + 16-s + 18-s + 3·19-s − 22-s − 7·23-s − 24-s + 26-s − 27-s − 8·29-s + 2·31-s + 32-s + 33-s + 36-s − 11·37-s + 3·38-s − 39-s + 11·41-s − 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.235·18-s + 0.688·19-s − 0.213·22-s − 1.45·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 1.48·29-s + 0.359·31-s + 0.176·32-s + 0.174·33-s + 1/6·36-s − 1.80·37-s + 0.486·38-s − 0.160·39-s + 1.71·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7350} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.40898777100439334409644388322, −6.77991783247166088382324103882, −5.90990953628545191507635906494, −5.56384347230256410591635588115, −4.77830304654899743253801740978, −3.97550366709580240426419923342, −3.35265102712246050292838611890, −2.27441409919212233727200164013, −1.41477069654366108263624274267, 0, 1.41477069654366108263624274267, 2.27441409919212233727200164013, 3.35265102712246050292838611890, 3.97550366709580240426419923342, 4.77830304654899743253801740978, 5.56384347230256410591635588115, 5.90990953628545191507635906494, 6.77991783247166088382324103882, 7.40898777100439334409644388322

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