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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 11-s + 2·13-s − 2·17-s + 8·23-s − 25-s + 6·29-s − 8·31-s + 6·37-s − 2·41-s − 8·47-s − 6·53-s − 2·55-s + 4·59-s − 6·61-s − 4·65-s + 4·67-s + 14·73-s + 4·79-s − 12·83-s + 4·85-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 1.16·47-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 1.63·73-s + 0.450·79-s − 1.31·83-s + 0.433·85-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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  =  1
Selberg data  =  $(2,\ 77616,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.37359126406776, −13.69467510822570, −13.31168896764603, −12.60331925944546, −12.48611511555815, −11.61592359386579, −11.23960435057659, −11.02788170655081, −10.37573893895687, −9.589107737037921, −9.298588032516067, −8.531640469877918, −8.321750575987015, −7.616330696774009, −7.120038147034913, −6.608184818294366, −6.107441291105620, −5.325506122645638, −4.785075694495303, −4.258794806091169, −3.599754675031618, −3.187523071190326, −2.431858954388236, −1.561781723360506, −0.8757656668354690, 0, 0.8757656668354690, 1.561781723360506, 2.431858954388236, 3.187523071190326, 3.599754675031618, 4.258794806091169, 4.785075694495303, 5.325506122645638, 6.107441291105620, 6.608184818294366, 7.120038147034913, 7.616330696774009, 8.321750575987015, 8.531640469877918, 9.298588032516067, 9.589107737037921, 10.37573893895687, 11.02788170655081, 11.23960435057659, 11.61592359386579, 12.48611511555815, 12.60331925944546, 13.31168896764603, 13.69467510822570, 14.37359126406776

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