Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 \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  − 2-s − 4-s − 7-s + 3·8-s + 6·13-s + 14-s − 16-s − 17-s + 4·19-s − 6·26-s + 28-s + 2·29-s − 5·32-s + 34-s + 6·37-s − 4·38-s + 10·41-s − 4·43-s + 4·47-s + 49-s − 6·52-s + 6·53-s − 3·56-s − 2·58-s − 4·59-s − 14·61-s + 7·64-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.242·17-s + 0.917·19-s − 1.17·26-s + 0.188·28-s + 0.371·29-s − 0.883·32-s + 0.171·34-s + 0.986·37-s − 0.648·38-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.832·52-s + 0.824·53-s − 0.400·56-s − 0.262·58-s − 0.520·59-s − 1.79·61-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(26775\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{26775} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 26775,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.538967581\)
\(L(\frac12)\)  \(\approx\)  \(1.538967581\)
\(L(\frac{3}{2})\)   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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.54434298902153, −14.67853980738882, −14.10820992527882, −13.64297179556321, −13.19961576714534, −12.78208818746297, −11.97731998734299, −11.38599133642863, −10.80562425249920, −10.39225615775210, −9.742463838858105, −9.099881353356766, −8.924745735735440, −8.148592302030026, −7.692916177321252, −7.095444803984390, −6.192033096242715, −5.905586125881592, −4.998415029591467, −4.371743122581391, −3.715653268168639, −3.130598489883633, −2.113188549316426, −1.160396809737752, −0.6622359251965893, 0.6622359251965893, 1.160396809737752, 2.113188549316426, 3.130598489883633, 3.715653268168639, 4.371743122581391, 4.998415029591467, 5.905586125881592, 6.192033096242715, 7.095444803984390, 7.692916177321252, 8.148592302030026, 8.924745735735440, 9.099881353356766, 9.742463838858105, 10.39225615775210, 10.80562425249920, 11.38599133642863, 11.97731998734299, 12.78208818746297, 13.19961576714534, 13.64297179556321, 14.10820992527882, 14.67853980738882, 15.54434298902153

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