Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11 \cdot 13^{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 + 4-s − 2·7-s − 8-s − 11-s + 2·14-s + 16-s + 6·17-s + 4·19-s + 22-s − 6·23-s − 5·25-s − 2·28-s − 6·29-s − 8·31-s − 32-s − 6·34-s + 10·37-s − 4·38-s + 6·41-s + 8·43-s − 44-s + 6·46-s − 6·47-s − 3·49-s + 5·50-s + 2·56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.301·11-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.213·22-s − 1.25·23-s − 25-s − 0.377·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.267·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33462\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33462} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 33462,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.001454397$
$L(\frac12)$  $\approx$  $1.001454397$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
11 \( 1 + T \)
13 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−15.04363582000027, −14.44754818358316, −14.12484746063238, −13.31514951113640, −12.82264002874983, −12.36172567474429, −11.78532070889138, −11.17955235529549, −10.78985925881790, −9.799909255033646, −9.703612732069229, −9.432571788426515, −8.442570347209899, −7.785493960601775, −7.619869647862185, −6.946883220601598, −6.021049346661547, −5.798455937185703, −5.193982644642716, −4.008613540331017, −3.649741420598570, −2.873144174058457, −2.173884432040065, −1.340373420147414, −0.4373436428150299, 0.4373436428150299, 1.340373420147414, 2.173884432040065, 2.873144174058457, 3.649741420598570, 4.008613540331017, 5.193982644642716, 5.798455937185703, 6.021049346661547, 6.946883220601598, 7.619869647862185, 7.785493960601775, 8.442570347209899, 9.432571788426515, 9.703612732069229, 9.799909255033646, 10.78985925881790, 11.17955235529549, 11.78532070889138, 12.36172567474429, 12.82264002874983, 13.31514951113640, 14.12484746063238, 14.44754818358316, 15.04363582000027

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