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Properties

Label 2-87120-1.1-c1-0-127
Degree $2$
Conductor $87120$
Sign $-1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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 + 7-s + 2·13-s − 6·17-s + 7·19-s − 6·23-s + 25-s + 31-s + 35-s − 7·37-s + 6·41-s − 8·43-s − 6·49-s + 6·53-s − 12·59-s − 61-s + 2·65-s + 7·67-s + 6·71-s − 13·73-s − 11·79-s − 6·85-s + 18·89-s + 2·91-s + 7·95-s − 97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.554·13-s − 1.45·17-s + 1.60·19-s − 1.25·23-s + 1/5·25-s + 0.179·31-s + 0.169·35-s − 1.15·37-s + 0.937·41-s − 1.21·43-s − 6/7·49-s + 0.824·53-s − 1.56·59-s − 0.128·61-s + 0.248·65-s + 0.855·67-s + 0.712·71-s − 1.52·73-s − 1.23·79-s − 0.650·85-s + 1.90·89-s + 0.209·91-s + 0.718·95-s − 0.101·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ -1)\)

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 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.05308374386334, −13.71567731055147, −13.25630429413952, −12.76792517169941, −12.07818690686754, −11.60097983514778, −11.32695930161325, −10.58019339528930, −10.26330082906368, −9.603887487196582, −9.182072602163165, −8.628296945821067, −8.165597528193918, −7.558081053757394, −7.027570926090892, −6.436883533391383, −5.937821259987000, −5.416148409466672, −4.754742712878773, −4.341016735140199, −3.513571500452013, −3.075821642577322, −2.144969920264614, −1.766749700055813, −0.9688743473395380, 0, 0.9688743473395380, 1.766749700055813, 2.144969920264614, 3.075821642577322, 3.513571500452013, 4.341016735140199, 4.754742712878773, 5.416148409466672, 5.937821259987000, 6.436883533391383, 7.027570926090892, 7.558081053757394, 8.165597528193918, 8.628296945821067, 9.182072602163165, 9.603887487196582, 10.26330082906368, 10.58019339528930, 11.32695930161325, 11.60097983514778, 12.07818690686754, 12.76792517169941, 13.25630429413952, 13.71567731055147, 14.05308374386334

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