Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s + 9-s + 2·11-s − 2·12-s + 5·13-s + 15-s + 4·16-s − 17-s − 2·19-s − 2·20-s − 23-s + 25-s + 27-s + 8·29-s − 31-s + 2·33-s − 2·36-s − 3·37-s + 5·39-s + 7·41-s − 4·44-s + 45-s + 47-s + 4·48-s − 51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.242·17-s − 0.458·19-s − 0.447·20-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.179·31-s + 0.348·33-s − 1/3·36-s − 0.493·37-s + 0.800·39-s + 1.09·41-s − 0.603·44-s + 0.149·45-s + 0.145·47-s + 0.577·48-s − 0.140·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12495\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{12495} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 12495,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.704240541\)
\(L(\frac12)\)  \(\approx\)  \(2.704240541\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−16.15760007704172, −15.83681377952675, −15.06843194399449, −14.37063594413095, −14.07618380046769, −13.61373233776650, −12.97259194480494, −12.61432425472013, −11.82476266053435, −11.03292486671603, −10.44970407164484, −9.851864209745338, −9.205380013756338, −8.779220340176560, −8.308188954849410, −7.695284841057807, −6.613503265695286, −6.273562652216530, −5.424601116616399, −4.675024751622071, −3.983451069266502, −3.507022464009211, −2.562181358321603, −1.552528734472064, −0.7868466074647328, 0.7868466074647328, 1.552528734472064, 2.562181358321603, 3.507022464009211, 3.983451069266502, 4.675024751622071, 5.424601116616399, 6.273562652216530, 6.613503265695286, 7.695284841057807, 8.308188954849410, 8.779220340176560, 9.205380013756338, 9.851864209745338, 10.44970407164484, 11.03292486671603, 11.82476266053435, 12.61432425472013, 12.97259194480494, 13.61373233776650, 14.07618380046769, 14.37063594413095, 15.06843194399449, 15.83681377952675, 16.15760007704172

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