Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 11-s − 6·13-s + 2·17-s − 4·19-s + 2·29-s − 8·31-s − 6·37-s − 10·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s + 4·59-s − 10·61-s − 12·67-s − 2·73-s − 77-s − 16·79-s − 4·83-s − 18·89-s − 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.301·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.371·29-s − 1.43·31-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 1.46·67-s − 0.234·73-s − 0.113·77-s − 1.80·79-s − 0.439·83-s − 1.90·89-s − 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(277200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{277200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 277200,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.12192902039933, −12.68631138912670, −12.30477131530438, −11.85962149858458, −11.53404626194666, −10.75357903620761, −10.48768680244093, −10.03641049364741, −9.645031734738987, −8.944750249940640, −8.611500424207869, −8.160957765178828, −7.458386511071523, −7.182237564838899, −6.865942499745568, −6.017828880905486, −5.542618940343627, −5.173649979030196, −4.558248867030589, −4.236216375209164, −3.430700055437628, −2.964648136741214, −2.300657223113676, −1.854731066131243, −1.222260660498716, 0, 0, 1.222260660498716, 1.854731066131243, 2.300657223113676, 2.964648136741214, 3.430700055437628, 4.236216375209164, 4.558248867030589, 5.173649979030196, 5.542618940343627, 6.017828880905486, 6.865942499745568, 7.182237564838899, 7.458386511071523, 8.160957765178828, 8.611500424207869, 8.944750249940640, 9.645031734738987, 10.03641049364741, 10.48768680244093, 10.75357903620761, 11.53404626194666, 11.85962149858458, 12.30477131530438, 12.68631138912670, 13.12192902039933

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