Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 7-s + 8-s + 11-s + 14-s − 16-s − 22-s − 23-s + 29-s + 37-s + 43-s + 46-s + 49-s + 2·53-s − 56-s − 58-s + 64-s + 67-s + 71-s − 74-s − 77-s − 79-s − 86-s + 88-s − 98-s − 2·106-s + 2·107-s + ⋯
L(s)  = 1  − 2-s − 7-s + 8-s + 11-s + 14-s − 16-s − 22-s − 23-s + 29-s + 37-s + 43-s + 46-s + 49-s + 2·53-s − 56-s − 58-s + 64-s + 67-s + 71-s − 74-s − 77-s − 79-s − 86-s + 88-s − 98-s − 2·106-s + 2·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1575} (1126, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5590472548\)
\(L(\frac12)\)  \(\approx\)  \(0.5590472548\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.623381387884658241348518171947, −8.910971909796422676638143030997, −8.247685610421616020023026468256, −7.30748434746590630575149047472, −6.58671895139458492128757551006, −5.74531244597510999207877374089, −4.43304395166399631448140218903, −3.71821822809850346474350049207, −2.36241087978402030020583646858, −0.929678056933204654072478137645, 0.929678056933204654072478137645, 2.36241087978402030020583646858, 3.71821822809850346474350049207, 4.43304395166399631448140218903, 5.74531244597510999207877374089, 6.58671895139458492128757551006, 7.30748434746590630575149047472, 8.247685610421616020023026468256, 8.910971909796422676638143030997, 9.623381387884658241348518171947

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