Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{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 + 4-s + 5-s + 7-s + 8-s + 10-s − 4·11-s + 14-s + 16-s − 2·17-s − 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s + 28-s + 2·29-s + 32-s − 2·34-s + 35-s − 6·37-s − 4·38-s + 40-s − 6·41-s − 4·43-s − 4·44-s + 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.188·28-s + 0.371·29-s + 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.986·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(106470\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{106470} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 106470,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.76438855793526, −13.33582001356346, −13.09881443043550, −12.69853921040286, −11.96281577976642, −11.60250068653877, −10.92416986090205, −10.61035827853865, −10.20504451113655, −9.629480607845962, −8.752030713047486, −8.580693280412359, −7.978293948872226, −7.273794095515420, −6.718610182360255, −6.552583959937562, −5.522014284740310, −5.207786501925213, −4.982256678930793, −4.101740116772765, −3.666995709835849, −2.684358590091952, −2.551024368279887, −1.780948816338996, −1.012908570585753, 0, 1.012908570585753, 1.780948816338996, 2.551024368279887, 2.684358590091952, 3.666995709835849, 4.101740116772765, 4.982256678930793, 5.207786501925213, 5.522014284740310, 6.552583959937562, 6.718610182360255, 7.273794095515420, 7.978293948872226, 8.580693280412359, 8.752030713047486, 9.629480607845962, 10.20504451113655, 10.61035827853865, 10.92416986090205, 11.60250068653877, 11.96281577976642, 12.69853921040286, 13.09881443043550, 13.33582001356346, 13.76438855793526

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