Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{2} \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  − 11-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 − 8·61-s + 4·67-s + 6·71-s − 2·73-s − 14·79-s + 12·83-s − 6·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.301·11-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 − 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 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(77616\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{77616} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 77616,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.364752297$
$L(\frac12)$  $\approx$  $1.364752297$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−13.75185417332295, −13.55818252445677, −13.16253784767267, −12.61370368470812, −12.03795062612613, −11.41681076568904, −11.00598464828802, −10.64297280613884, −10.13016838546717, −9.280986447851605, −9.052113106885213, −8.394601263805273, −8.096504655623774, −7.306599111906224, −6.678797570755795, −6.450246048067936, −5.661936233494797, −5.230842715111594, −4.376271847366752, −4.104365130023346, −3.342391196139998, −2.684983039328964, −1.991335468970794, −1.390866676776805, −0.3774879598429732, 0.3774879598429732, 1.390866676776805, 1.991335468970794, 2.684983039328964, 3.342391196139998, 4.104365130023346, 4.376271847366752, 5.230842715111594, 5.661936233494797, 6.450246048067936, 6.678797570755795, 7.306599111906224, 8.096504655623774, 8.394601263805273, 9.052113106885213, 9.280986447851605, 10.13016838546717, 10.64297280613884, 11.00598464828802, 11.41681076568904, 12.03795062612613, 12.61370368470812, 13.16253784767267, 13.55818252445677, 13.75185417332295

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