Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 11^{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  + 2·7-s − 4·13-s − 6·17-s + 4·19-s + 6·23-s − 5·25-s − 6·29-s − 8·31-s + 10·37-s + 6·41-s − 8·43-s − 6·47-s − 3·49-s + 8·61-s − 4·67-s + 6·71-s − 2·73-s + 14·79-s − 12·83-s + 6·89-s − 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 1.02·61-s − 0.488·67-s + 0.712·71-s − 0.234·73-s + 1.57·79-s − 1.31·83-s + 0.635·89-s − 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{69696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 69696,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.539648238$
$L(\frac12)$  $\approx$  $1.539648238$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
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

−14.34298029913780, −13.40045370369046, −13.27619782476649, −12.76083848014674, −12.06021157064686, −11.50941065750706, −11.14454753549008, −10.88717544223741, −9.974264855501646, −9.442216354559777, −9.260278553722083, −8.502249775033518, −7.867782859184465, −7.494274331618060, −6.993542584772059, −6.397527940669974, −5.628183201748769, −5.126687024172126, −4.712166133058094, −4.057019653466659, −3.408658410016388, −2.593138061351588, −2.071553047636901, −1.425596413659813, −0.4040665185483156, 0.4040665185483156, 1.425596413659813, 2.071553047636901, 2.593138061351588, 3.408658410016388, 4.057019653466659, 4.712166133058094, 5.126687024172126, 5.628183201748769, 6.397527940669974, 6.993542584772059, 7.494274331618060, 7.867782859184465, 8.502249775033518, 9.260278553722083, 9.442216354559777, 9.974264855501646, 10.88717544223741, 11.14454753549008, 11.50941065750706, 12.06021157064686, 12.76083848014674, 13.27619782476649, 13.40045370369046, 14.34298029913780

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