Properties

Degree $2$
Conductor $35520$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s − 4·19-s − 8·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s − 37-s + 2·39-s + 10·41-s − 12·43-s − 45-s − 7·49-s + 2·51-s − 6·53-s + 4·55-s − 4·57-s − 4·59-s + 10·61-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.164·37-s + 0.320·39-s + 1.56·41-s − 1.82·43-s − 0.149·45-s − 49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35520\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 37\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{35520} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.726997099\)
\(L(\frac12)\) \(\approx\) \(1.726997099\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
37 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.88130991670117, −14.40174169361483, −13.95322705430600, −13.26873267402489, −12.97874894163842, −12.34606977647038, −11.81650574815274, −11.24927422384640, −10.55126601424679, −10.16346128469609, −9.716954114841370, −8.903985884773862, −8.256626643449856, −8.011426298834883, −7.637755079025635, −6.599083957424478, −6.329240577421901, −5.486631271210067, −4.821363251826552, −4.212741589696033, −3.614643601240872, −2.910802693943082, −2.321450390744012, −1.536849356875348, −0.4640197244987523, 0.4640197244987523, 1.536849356875348, 2.321450390744012, 2.910802693943082, 3.614643601240872, 4.212741589696033, 4.821363251826552, 5.486631271210067, 6.329240577421901, 6.599083957424478, 7.637755079025635, 8.011426298834883, 8.256626643449856, 8.903985884773862, 9.716954114841370, 10.16346128469609, 10.55126601424679, 11.24927422384640, 11.81650574815274, 12.34606977647038, 12.97874894163842, 13.26873267402489, 13.95322705430600, 14.40174169361483, 14.88130991670117

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