Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 6·13-s + 2·17-s − 8·19-s − 8·23-s + 25-s − 2·29-s − 4·31-s + 2·37-s − 6·41-s − 4·43-s + 8·47-s + 10·53-s − 4·59-s − 2·61-s + 6·65-s − 4·67-s + 12·71-s + 2·73-s + 8·79-s + 4·83-s − 2·85-s − 6·89-s + 8·95-s + 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1.37·53-s − 0.520·59-s − 0.256·61-s + 0.744·65-s − 0.488·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s − 0.635·89-s + 0.820·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{141120} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 141120,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.56982569229651, −13.17653889249694, −12.45248316882646, −12.24236894521453, −11.94172767358865, −11.23960056741507, −10.74653808081196, −10.13401007203127, −10.01257682643970, −9.259208748508762, −8.790467534302801, −8.232720236788009, −7.715174506425591, −7.418472309023552, −6.727718847133253, −6.306813069332471, −5.641871862100168, −5.119866479272172, −4.555653219727702, −4.000379375422736, −3.617842645803157, −2.739513445635640, −2.159900468817440, −1.806237177620919, −0.5924975365469867, 0, 0.5924975365469867, 1.806237177620919, 2.159900468817440, 2.739513445635640, 3.617842645803157, 4.000379375422736, 4.555653219727702, 5.119866479272172, 5.641871862100168, 6.306813069332471, 6.727718847133253, 7.418472309023552, 7.715174506425591, 8.232720236788009, 8.790467534302801, 9.259208748508762, 10.01257682643970, 10.13401007203127, 10.74653808081196, 11.23960056741507, 11.94172767358865, 12.24236894521453, 12.45248316882646, 13.17653889249694, 13.56982569229651

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