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Properties

Label 2-18097-1.1-c1-0-0
Degree $2$
Conductor $18097$
Sign $-1$
Analytic cond. $144.505$
Root an. cond. $12.0210$
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-s − 3-s − 4-s − 2·5-s + 6-s − 7-s + 3·8-s − 2·9-s + 2·10-s + 6·11-s + 12-s + 3·13-s + 14-s + 2·15-s − 16-s − 3·17-s + 2·18-s + 4·19-s + 2·20-s + 21-s − 6·22-s + 8·23-s − 3·24-s − 25-s − 3·26-s + 5·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 1.80·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s − 1.27·22-s + 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.588·26-s + 0.962·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18097\)
Sign: $-1$
Analytic conductor: \(144.505\)
Root analytic conductor: \(12.0210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 18097,\ (\ :1/2),\ -1)\)

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)$
bad18097 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−16.19884588678809, −15.78492239938360, −14.99537622059829, −14.37072932104886, −13.96176449640991, −13.34027101133937, −12.63505195697111, −12.06431627729623, −11.52161163287565, −10.98264896467496, −10.76668004953125, −9.628455520050322, −9.286720785500189, −8.767842339963209, −8.324539447283898, −7.550857241522824, −6.876529926455985, −6.397128687021721, −5.679425334128508, −4.768089617074839, −4.345224944790220, −3.534267937974372, −3.029697676422826, −1.477206311389783, −0.9214031251721340, 0, 0.9214031251721340, 1.477206311389783, 3.029697676422826, 3.534267937974372, 4.345224944790220, 4.768089617074839, 5.679425334128508, 6.397128687021721, 6.876529926455985, 7.550857241522824, 8.324539447283898, 8.767842339963209, 9.286720785500189, 9.628455520050322, 10.76668004953125, 10.98264896467496, 11.52161163287565, 12.06431627729623, 12.63505195697111, 13.34027101133937, 13.96176449640991, 14.37072932104886, 14.99537622059829, 15.78492239938360, 16.19884588678809

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