Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
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 + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s + 4·13-s − 14-s + 16-s + 4·17-s + 8·19-s − 20-s − 22-s − 2·23-s + 25-s − 4·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 4·34-s − 35-s + 4·37-s − 8·38-s + 40-s − 2·41-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 + 0.301·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.417·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s + 0.657·37-s − 1.29·38-s + 0.158·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6930} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.752626305$
$L(\frac12)$  $\approx$  $1.752626305$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−17.30993985849517, −16.48583252075295, −16.07335024419811, −15.73230899658461, −14.89664302871985, −14.25099456705277, −13.82389230552656, −12.98712642786623, −12.11439638820719, −11.81330399469523, −11.13782373598985, −10.61211950320020, −9.767453990338751, −9.368977752898051, −8.426554136967762, −8.106528583424502, −7.366870052040051, −6.785578200127078, −5.818450825230832, −5.341275134150653, −4.214238363187212, −3.499620459725822, −2.741680689077122, −1.450192465560854, −0.8506690019308983, 0.8506690019308983, 1.450192465560854, 2.741680689077122, 3.499620459725822, 4.214238363187212, 5.341275134150653, 5.818450825230832, 6.785578200127078, 7.366870052040051, 8.106528583424502, 8.426554136967762, 9.368977752898051, 9.767453990338751, 10.61211950320020, 11.13782373598985, 11.81330399469523, 12.11439638820719, 12.98712642786623, 13.82389230552656, 14.25099456705277, 14.89664302871985, 15.73230899658461, 16.07335024419811, 16.48583252075295, 17.30993985849517

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