Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
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 + 7-s + 3·8-s − 4·11-s + 2·13-s − 14-s − 16-s − 6·17-s + 4·19-s + 4·22-s − 2·26-s − 28-s + 2·29-s − 5·32-s + 6·34-s − 6·37-s − 4·38-s − 2·41-s + 4·43-s + 4·44-s + 49-s − 2·52-s + 6·53-s + 3·56-s − 2·58-s − 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.852·22-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.400·56-s − 0.262·58-s − 1.56·59-s + ⋯

Functional equation

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

Invariants

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

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.50122194471590, −18.53383007714365, −18.36186827960758, −17.63027013104801, −17.20125444420117, −16.12928822969394, −15.77812343202871, −14.96850797077115, −13.87790413706979, −13.58664774117488, −12.88523910328498, −11.95519528934951, −10.92750490716523, −10.61897832346732, −9.717363147944347, −8.941556964198447, −8.352294990335304, −7.654943630241817, −6.853608128702849, −5.615335303903544, −4.879836468908928, −4.037969968796789, −2.727753342830548, −1.460361501877125, 0, 1.460361501877125, 2.727753342830548, 4.037969968796789, 4.879836468908928, 5.615335303903544, 6.853608128702849, 7.654943630241817, 8.352294990335304, 8.941556964198447, 9.717363147944347, 10.61897832346732, 10.92750490716523, 11.95519528934951, 12.88523910328498, 13.58664774117488, 13.87790413706979, 14.96850797077115, 15.77812343202871, 16.12928822969394, 17.20125444420117, 17.63027013104801, 18.36186827960758, 18.53383007714365, 19.50122194471590

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