Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 \cdot 17^{2} $
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 − 5-s − 7-s − 3·8-s − 10-s − 6·13-s − 14-s − 16-s + 4·19-s + 20-s + 25-s − 6·26-s + 28-s − 2·29-s + 5·32-s + 35-s + 6·37-s + 4·38-s + 3·40-s − 10·41-s + 4·43-s − 4·47-s + 49-s + 50-s + 6·52-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 0.316·10-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.223·20-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 0.371·29-s + 0.883·32-s + 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.474·40-s − 1.56·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.141·50-s + 0.832·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(91035\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{91035} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 91035,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.8346223549\)
\(L(\frac12)\)  \(\approx\)  \(0.8346223549\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
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

−13.88000937944593, −13.32420231170479, −12.88947689353230, −12.40300952222353, −12.05145707099593, −11.58028671581844, −11.08286351587631, −10.20604072326716, −9.876794857296465, −9.375553664830620, −9.024803619232381, −8.198429504631209, −7.821223917789506, −7.244401361935527, −6.678090052672340, −6.128574173307996, −5.395671954735817, −4.979002846899712, −4.648889982180631, −3.809623704222444, −3.485356036069146, −2.762384197811691, −2.290172217212825, −1.167753610786524, −0.2790988009244908, 0.2790988009244908, 1.167753610786524, 2.290172217212825, 2.762384197811691, 3.485356036069146, 3.809623704222444, 4.648889982180631, 4.979002846899712, 5.395671954735817, 6.128574173307996, 6.678090052672340, 7.244401361935527, 7.821223917789506, 8.198429504631209, 9.024803619232381, 9.375553664830620, 9.876794857296465, 10.20604072326716, 11.08286351587631, 11.58028671581844, 12.05145707099593, 12.40300952222353, 12.88947689353230, 13.32420231170479, 13.88000937944593

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