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  =  \((2,\ 1575,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.825475986\)
\(L(\frac12)\)  \(\approx\)  \(1.825475986\)
\(L(1)\)   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

−9.480324702670257276224191364595, −8.809514046795125395067500520923, −8.116836408344220554470497759149, −6.96970667009939880157183486674, −6.27022701614165737024812194838, −5.22857436838188553221763340267, −4.69504563308156006234533030342, −3.84468175199313457099304344543, −2.88846386044735407266517185951, −1.46488067501830059494637558784, 1.46488067501830059494637558784, 2.88846386044735407266517185951, 3.84468175199313457099304344543, 4.69504563308156006234533030342, 5.22857436838188553221763340267, 6.27022701614165737024812194838, 6.96970667009939880157183486674, 8.116836408344220554470497759149, 8.809514046795125395067500520923, 9.480324702670257276224191364595

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