Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 37^{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 − 3-s + 4-s + 2·5-s + 6-s − 2·7-s − 8-s + 9-s − 2·10-s − 11-s − 12-s + 6·13-s + 2·14-s − 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 2·21-s + 22-s + 6·23-s + 24-s − 25-s − 6·26-s − 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.66·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.436·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(90354\)    =    \(2 \cdot 3 \cdot 11 \cdot 37^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{90354} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 90354,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.825062167\)
\(L(\frac12)\)  \(\approx\)  \(1.825062167\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;37\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 + T \)
37 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.83640794360996, −13.26700094224258, −12.88601126742939, −12.51319051350298, −11.62046418864760, −11.36310568105941, −10.80021929030938, −10.40924947369576, −9.798375612693698, −9.523850853399223, −8.889239009750564, −8.510105420451083, −7.862398775473569, −7.086822389499400, −6.750213313739091, −6.219778506816490, −5.724670479360805, −5.396199002092112, −4.518145575144424, −3.865695874482165, −3.043634532775068, −2.715229249379934, −1.730398671655385, −1.173128406881317, −0.5616137768686578, 0.5616137768686578, 1.173128406881317, 1.730398671655385, 2.715229249379934, 3.043634532775068, 3.865695874482165, 4.518145575144424, 5.396199002092112, 5.724670479360805, 6.219778506816490, 6.750213313739091, 7.086822389499400, 7.862398775473569, 8.510105420451083, 8.889239009750564, 9.523850853399223, 9.798375612693698, 10.40924947369576, 10.80021929030938, 11.36310568105941, 11.62046418864760, 12.51319051350298, 12.88601126742939, 13.26700094224258, 13.83640794360996

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