Properties

Degree $2$
Conductor $55470$
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-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s + 12-s + 3·13-s − 14-s + 15-s + 16-s − 2·17-s − 18-s + 19-s + 20-s + 21-s + 2·22-s + 8·23-s − 24-s + 25-s − 3·26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55470\)    =    \(2 \cdot 3 \cdot 5 \cdot 43^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{55470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
43 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.83121577405986, −14.07596915645410, −13.54441452907195, −13.35171272884276, −12.65208766668613, −12.09031480467625, −11.45237080665135, −10.91858441631470, −10.47406299147989, −10.10756599375158, −9.310392941096838, −8.906856941425189, −8.576110004680457, −7.973220277361217, −7.409555265422383, −6.851976166176649, −6.365646397667336, −5.619474983211032, −5.024782688691927, −4.463918751943944, −3.547858816081124, −2.956709354104225, −2.463865463375233, −1.545600697596958, −1.188180629176832, 0, 1.188180629176832, 1.545600697596958, 2.463865463375233, 2.956709354104225, 3.547858816081124, 4.463918751943944, 5.024782688691927, 5.619474983211032, 6.365646397667336, 6.851976166176649, 7.409555265422383, 7.973220277361217, 8.576110004680457, 8.906856941425189, 9.310392941096838, 10.10756599375158, 10.47406299147989, 10.91858441631470, 11.45237080665135, 12.09031480467625, 12.65208766668613, 13.35171272884276, 13.54441452907195, 14.07596915645410, 14.83121577405986

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