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 + 3·5-s + 6-s − 8-s + 9-s − 3·10-s + 11-s − 12-s + 4·13-s − 3·15-s + 16-s − 5·17-s − 18-s − 8·19-s + 3·20-s − 22-s + 6·23-s + 24-s + 4·25-s − 4·26-s − 27-s + 4·29-s + 3·30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.774·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 1.83·19-s + 0.670·20-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.784·26-s − 0.192·27-s + 0.742·29-s + 0.547·30-s + 0.359·31-s − 0.176·32-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  =  \((2,\ 90354,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.829959072\)
\(L(\frac12)\)  \(\approx\)  \(1.829959072\)
\(L(\frac{3}{2})\)   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 \( 1 - 3 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.59067216781052, −13.38328524191941, −12.94293114123491, −12.47029326710368, −11.68028752428032, −11.24232636737372, −10.80785910663333, −10.40286394283624, −10.00468558956832, −9.238438370708463, −9.001601964140761, −8.467665979817446, −8.016518917667956, −6.986779483966989, −6.626872793915662, −6.280867887995219, −5.955795251864472, −5.062230914545232, −4.693075206580036, −3.919555553636384, −3.160258457469755, −2.319967369414888, −1.920339512137433, −1.263614183200874, −0.5210851827350906, 0.5210851827350906, 1.263614183200874, 1.920339512137433, 2.319967369414888, 3.160258457469755, 3.919555553636384, 4.693075206580036, 5.062230914545232, 5.955795251864472, 6.280867887995219, 6.626872793915662, 6.986779483966989, 8.016518917667956, 8.467665979817446, 9.001601964140761, 9.238438370708463, 10.00468558956832, 10.40286394283624, 10.80785910663333, 11.24232636737372, 11.68028752428032, 12.47029326710368, 12.94293114123491, 13.38328524191941, 13.59067216781052

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