Properties

Label 2-55470-1.1-c1-0-33
Degree $2$
Conductor $55470$
Sign $1$
Analytic cond. $442.930$
Root an. cond. $21.0459$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

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 − 5·11-s − 12-s + 4·13-s + 14-s + 15-s + 16-s + 2·17-s − 18-s − 6·19-s − 20-s + 21-s + 5·22-s + 6·23-s + 24-s + 25-s − 4·26-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 − 1.50·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-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: $1$
Analytic conductor: \(442.930\)
Root analytic conductor: \(21.0459\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.03933449182112, −14.72966608427871, −13.68841370778568, −13.19712321220026, −12.83721907866773, −12.47357222512488, −11.49788967801717, −11.41023489059745, −10.64319387602880, −10.43765313883405, −9.973760402419754, −9.055909265728482, −8.749035089367970, −8.138851084200000, −7.683994536918983, −6.910449038625806, −6.743994068156349, −5.811836935357164, −5.474008173418694, −4.844847097348249, −3.999117362731977, −3.381409782079104, −2.820908925945954, −1.888438429175949, −1.266765722791829, 0, 0, 1.266765722791829, 1.888438429175949, 2.820908925945954, 3.381409782079104, 3.999117362731977, 4.844847097348249, 5.474008173418694, 5.811836935357164, 6.743994068156349, 6.910449038625806, 7.683994536918983, 8.138851084200000, 8.749035089367970, 9.055909265728482, 9.973760402419754, 10.43765313883405, 10.64319387602880, 11.41023489059745, 11.49788967801717, 12.47357222512488, 12.83721907866773, 13.19712321220026, 13.68841370778568, 14.72966608427871, 15.03933449182112

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