Properties

Label 2-87120-1.1-c1-0-107
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 + 4·7-s − 6·13-s − 6·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s − 10·37-s + 10·41-s + 4·47-s + 9·49-s + 10·53-s − 4·59-s + 2·61-s + 6·65-s + 8·67-s + 14·73-s − 16·79-s + 8·83-s + 6·85-s + 6·89-s − 24·91-s − 4·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 1.66·13-s − 1.45·17-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 − 1.64·37-s + 1.56·41-s + 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.744·65-s + 0.977·67-s + 1.63·73-s − 1.80·79-s + 0.878·83-s + 0.650·85-s + 0.635·89-s − 2.51·91-s − 0.410·95-s + 0.203·97-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.26353958079097, −13.75201804076449, −13.16861579554792, −12.51553954984180, −12.19847179732289, −11.59871374326656, −11.21661194363550, −10.78579663406048, −10.36886319338247, −9.413739320486630, −9.221400140687911, −8.632716255782946, −7.989245745590448, −7.592770213602847, −6.981936648473513, −6.897756234485526, −5.631346158362721, −5.280945833348139, −4.867540026592487, −4.267361218294928, −3.784170560696384, −2.850464185656171, −2.263019297968219, −1.769932433114861, −0.8690978881654730, 0, 0.8690978881654730, 1.769932433114861, 2.263019297968219, 2.850464185656171, 3.784170560696384, 4.267361218294928, 4.867540026592487, 5.280945833348139, 5.631346158362721, 6.897756234485526, 6.981936648473513, 7.592770213602847, 7.989245745590448, 8.632716255782946, 9.221400140687911, 9.413739320486630, 10.36886319338247, 10.78579663406048, 11.21661194363550, 11.59871374326656, 12.19847179732289, 12.51553954984180, 13.16861579554792, 13.75201804076449, 14.26353958079097

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