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Properties

Label 2-70560-1.1-c1-0-13
Degree $2$
Conductor $70560$
Sign $1$
Analytic cond. $563.424$
Root an. cond. $23.7365$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·11-s − 6·13-s − 6·17-s + 4·19-s − 8·23-s + 25-s − 10·29-s + 4·31-s − 6·37-s + 6·41-s − 4·43-s + 12·47-s − 6·53-s + 4·55-s − 4·59-s + 2·61-s − 6·65-s − 4·67-s + 2·73-s + 8·79-s + 12·83-s − 6·85-s + 14·89-s + 4·95-s − 6·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.744·65-s − 0.488·67-s + 0.234·73-s + 0.900·79-s + 1.31·83-s − 0.650·85-s + 1.48·89-s + 0.410·95-s − 0.609·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70560\)    =    \(2^{5} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(563.424\)
Root analytic conductor: \(23.7365\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 70560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.440598933\)
\(L(\frac12)\) \(\approx\) \(1.440598933\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.13938175642733, −13.65589239687018, −13.29831772675477, −12.38493708441374, −12.23329458237024, −11.71436067427161, −11.16260914777333, −10.58510049321086, −9.982942618133506, −9.442134011009782, −9.260509188104355, −8.662329815905223, −7.828235841670444, −7.423269910755853, −6.911160697239772, −6.280706607030459, −5.871273262582651, −5.112320868454908, −4.649532081072183, −3.972913298892186, −3.503898315732992, −2.461852988465378, −2.148108088316679, −1.449920118776275, −0.3833155250957422, 0.3833155250957422, 1.449920118776275, 2.148108088316679, 2.461852988465378, 3.503898315732992, 3.972913298892186, 4.649532081072183, 5.112320868454908, 5.871273262582651, 6.280706607030459, 6.911160697239772, 7.423269910755853, 7.828235841670444, 8.662329815905223, 9.260509188104355, 9.442134011009782, 9.982942618133506, 10.58510049321086, 11.16260914777333, 11.71436067427161, 12.23329458237024, 12.38493708441374, 13.29831772675477, 13.65589239687018, 14.13938175642733

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