Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 37^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 5-s − 6-s + 7-s + 3·8-s + 9-s + 10-s − 12-s + 6·13-s − 14-s − 15-s − 16-s − 2·17-s − 18-s + 8·19-s + 20-s + 21-s − 8·23-s + 3·24-s + 25-s − 6·26-s + 27-s − 28-s + 2·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s − 1.66·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 143745 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 143745 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(143745\)    =    \(3 \cdot 5 \cdot 7 \cdot 37^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{143745} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 143745,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7,\;37\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
37 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.66558236853121, −13.31793768763056, −12.81622743753099, −11.96232931894768, −11.75486842013252, −11.20153968202952, −10.51845242457393, −10.28281693730323, −9.701472600749713, −9.113237513560223, −8.755188729936654, −8.370761469747322, −7.916764080024939, −7.457761618677973, −7.022831438001295, −6.247044041547369, −5.626450994797891, −5.153399666143463, −4.409628005460023, −3.906130527286880, −3.609273283889312, −2.891277505922253, −1.991748417510131, −1.411074173873274, −0.9125523225967206, 0, 0.9125523225967206, 1.411074173873274, 1.991748417510131, 2.891277505922253, 3.609273283889312, 3.906130527286880, 4.409628005460023, 5.153399666143463, 5.626450994797891, 6.247044041547369, 7.022831438001295, 7.457761618677973, 7.916764080024939, 8.370761469747322, 8.755188729936654, 9.113237513560223, 9.701472600749713, 10.28281693730323, 10.51845242457393, 11.20153968202952, 11.75486842013252, 11.96232931894768, 12.81622743753099, 13.31793768763056, 13.66558236853121

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