Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 17 $
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 + 2·7-s + 8-s − 10-s + 4·13-s + 2·14-s + 16-s + 17-s + 6·19-s − 20-s + 2·23-s + 25-s + 4·26-s + 2·28-s − 6·29-s − 4·31-s + 32-s + 34-s − 2·35-s − 2·37-s + 6·38-s − 40-s − 6·41-s − 6·43-s + 2·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.223·20-s + 0.417·23-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s − 0.328·37-s + 0.973·38-s − 0.158·40-s − 0.937·41-s − 0.914·43-s + 0.294·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(185130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{185130} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 185130,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.43874777606920, −12.97335660050723, −12.40038735940879, −11.90063947223908, −11.57065228444614, −11.08388815805408, −10.80846798463398, −10.19795557290444, −9.588411730821816, −9.003865256321245, −8.583242121315663, −7.998192990219636, −7.553780755476860, −7.133978181952863, −6.638861901099861, −5.822982131340177, −5.569278724756641, −5.047695592438388, −4.479682721420056, −3.910600906709149, −3.374513349433521, −3.071554674615800, −2.145102648614548, −1.483746544062067, −1.093078599282872, 0, 1.093078599282872, 1.483746544062067, 2.145102648614548, 3.071554674615800, 3.374513349433521, 3.910600906709149, 4.479682721420056, 5.047695592438388, 5.569278724756641, 5.822982131340177, 6.638861901099861, 7.133978181952863, 7.553780755476860, 7.998192990219636, 8.583242121315663, 9.003865256321245, 9.588411730821816, 10.19795557290444, 10.80846798463398, 11.08388815805408, 11.57065228444614, 11.90063947223908, 12.40038735940879, 12.97335660050723, 13.43874777606920

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