Properties

Label 2-107712-1.1-c1-0-81
Degree $2$
Conductor $107712$
Sign $-1$
Analytic cond. $860.084$
Root an. cond. $29.3271$
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  − 2·5-s + 4·7-s + 11-s + 4·13-s + 17-s − 8·19-s − 25-s − 10·31-s − 8·35-s − 8·37-s + 10·41-s − 8·43-s + 10·47-s + 9·49-s − 12·53-s − 2·55-s + 8·59-s + 2·61-s − 8·65-s + 4·67-s − 4·71-s + 10·73-s + 4·77-s − 12·79-s − 4·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 0.301·11-s + 1.10·13-s + 0.242·17-s − 1.83·19-s − 1/5·25-s − 1.79·31-s − 1.35·35-s − 1.31·37-s + 1.56·41-s − 1.21·43-s + 1.45·47-s + 9/7·49-s − 1.64·53-s − 0.269·55-s + 1.04·59-s + 0.256·61-s − 0.992·65-s + 0.488·67-s − 0.474·71-s + 1.17·73-s + 0.455·77-s − 1.35·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(107712\)    =    \(2^{6} \cdot 3^{2} \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(860.084\)
Root analytic conductor: \(29.3271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 107712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.23093587722528, −13.36024897145235, −12.95866380370763, −12.32377987488672, −12.01747994844592, −11.29803013955386, −11.00679768396496, −10.83516901644211, −10.16217602126119, −9.305200915612768, −8.854358471293256, −8.300226954641359, −8.159974208277645, −7.512292879574425, −6.982343622003507, −6.436053219274789, −5.654253163977611, −5.367647422170864, −4.489575722239879, −4.134446963169419, −3.769823751485002, −3.031123050963685, −1.918854241038231, −1.821846306173694, −0.8834935160505292, 0, 0.8834935160505292, 1.821846306173694, 1.918854241038231, 3.031123050963685, 3.769823751485002, 4.134446963169419, 4.489575722239879, 5.367647422170864, 5.654253163977611, 6.436053219274789, 6.982343622003507, 7.512292879574425, 8.159974208277645, 8.300226954641359, 8.854358471293256, 9.305200915612768, 10.16217602126119, 10.83516901644211, 11.00679768396496, 11.29803013955386, 12.01747994844592, 12.32377987488672, 12.95866380370763, 13.36024897145235, 14.23093587722528

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