Properties

Label 2-43560-1.1-c1-0-56
Degree $2$
Conductor $43560$
Sign $-1$
Analytic cond. $347.828$
Root an. cond. $18.6501$
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  − 5-s + 4·7-s + 2·13-s + 2·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s + 6·37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s + 4·59-s + 2·61-s − 2·65-s + 8·67-s + 6·73-s − 16·83-s − 2·85-s + 6·89-s + 8·91-s + 4·95-s − 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.702·73-s − 1.75·83-s − 0.216·85-s + 0.635·89-s + 0.838·91-s + 0.410·95-s − 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43560\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(347.828\)
Root analytic conductor: \(18.6501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43560,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.83024770922495, −14.47673237636014, −14.10259529571048, −13.40850051127192, −12.78538835313349, −12.39764274170998, −11.67730353800334, −11.28813755736317, −10.92268349597340, −10.40847266019430, −9.662637701150148, −9.081676908409405, −8.385526582652375, −8.105891802241959, −7.639302302410770, −6.989193532005891, −6.302968568930480, −5.587900928321149, −5.191496443141863, −4.352713102987741, −4.043040138408595, −3.333766459040656, −2.350831722555156, −1.773874121610584, −1.088448086118744, 0, 1.088448086118744, 1.773874121610584, 2.350831722555156, 3.333766459040656, 4.043040138408595, 4.352713102987741, 5.191496443141863, 5.587900928321149, 6.302968568930480, 6.989193532005891, 7.639302302410770, 8.105891802241959, 8.385526582652375, 9.081676908409405, 9.662637701150148, 10.40847266019430, 10.92268349597340, 11.28813755736317, 11.67730353800334, 12.39764274170998, 12.78538835313349, 13.40850051127192, 14.10259529571048, 14.47673237636014, 14.83024770922495

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