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Properties

Label 2-43120-1.1-c1-0-47
Degree $2$
Conductor $43120$
Sign $-1$
Analytic cond. $344.314$
Root an. cond. $18.5557$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 9-s + 11-s + 4·13-s − 2·15-s − 4·19-s + 25-s + 4·27-s − 6·29-s − 10·31-s − 2·33-s + 2·37-s − 8·39-s + 12·41-s + 4·43-s + 45-s + 6·47-s − 6·53-s + 55-s + 8·57-s − 6·59-s + 4·61-s + 4·65-s + 4·67-s − 12·71-s + 4·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.516·15-s − 0.917·19-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.79·31-s − 0.348·33-s + 0.328·37-s − 1.28·39-s + 1.87·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s − 0.824·53-s + 0.134·55-s + 1.05·57-s − 0.781·59-s + 0.512·61-s + 0.496·65-s + 0.488·67-s − 1.42·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43120\)    =    \(2^{4} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(344.314\)
Root analytic conductor: \(18.5557\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.85188259354199, −14.44271393221832, −13.98658197605038, −13.21019031880403, −12.77901252114533, −12.51809824587181, −11.68410942464997, −11.23810684556504, −10.78660652959646, −10.61161695764228, −9.707300068313819, −9.072338658355758, −8.863699708493036, −7.968627904037360, −7.388896742928497, −6.717518040718592, −6.163897740689768, −5.741247812563959, −5.437319637648448, −4.480853078626842, −4.052261776343397, −3.306243465171841, −2.391788386826634, −1.655013454284535, −0.9042475328643629, 0, 0.9042475328643629, 1.655013454284535, 2.391788386826634, 3.306243465171841, 4.052261776343397, 4.480853078626842, 5.437319637648448, 5.741247812563959, 6.163897740689768, 6.717518040718592, 7.388896742928497, 7.968627904037360, 8.863699708493036, 9.072338658355758, 9.707300068313819, 10.61161695764228, 10.78660652959646, 11.23810684556504, 11.68410942464997, 12.51809824587181, 12.77901252114533, 13.21019031880403, 13.98658197605038, 14.44271393221832, 14.85188259354199

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