Properties

Degree 2
Conductor $ 2^{6} \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 + 6·13-s + 2·17-s − 4·19-s − 25-s − 2·29-s + 8·31-s − 6·37-s + 10·41-s − 4·43-s + 8·47-s + 6·53-s + 2·55-s + 4·59-s − 10·61-s + 12·65-s − 12·67-s − 2·73-s − 16·79-s + 4·83-s + 4·85-s + 18·89-s − 8·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 1/5·25-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 + 0.824·53-s + 0.269·55-s + 0.520·59-s − 1.28·61-s + 1.48·65-s − 1.46·67-s − 0.234·73-s − 1.80·79-s + 0.439·83-s + 0.433·85-s + 1.90·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{310464} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 310464,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
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

−13.00817490723709, −12.43018447960806, −11.93787877955671, −11.59292198819247, −10.90936600001468, −10.55861039030530, −10.27678583119946, −9.657496292813573, −9.177960056631174, −8.766283618889881, −8.400313308353257, −7.861458399626094, −7.249651392387117, −6.739703976340073, −6.149618686546127, −5.866037784858511, −5.639201332078691, −4.676465421521772, −4.371272545008197, −3.678561849816847, −3.311726649414809, −2.505121367324529, −2.091984411180895, −1.303114660392459, −1.055181778177779, 0, 1.055181778177779, 1.303114660392459, 2.091984411180895, 2.505121367324529, 3.311726649414809, 3.678561849816847, 4.371272545008197, 4.676465421521772, 5.639201332078691, 5.866037784858511, 6.149618686546127, 6.739703976340073, 7.249651392387117, 7.861458399626094, 8.400313308353257, 8.766283618889881, 9.177960056631174, 9.657496292813573, 10.27678583119946, 10.55861039030530, 10.90936600001468, 11.59292198819247, 11.93787877955671, 12.43018447960806, 13.00817490723709

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