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Properties

Label 2-18032-1.1-c1-0-17
Degree $2$
Conductor $18032$
Sign $1$
Analytic cond. $143.986$
Root an. cond. $11.9994$
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  + 3·3-s + 6·9-s + 5·13-s + 6·17-s + 6·19-s − 23-s − 5·25-s + 9·27-s + 9·29-s + 3·31-s − 8·37-s + 15·39-s − 3·41-s + 8·43-s + 7·47-s + 18·51-s − 2·53-s + 18·57-s + 4·59-s + 10·61-s − 8·67-s − 3·69-s − 7·71-s − 9·73-s − 15·75-s + 6·79-s + 9·81-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s + 1.38·13-s + 1.45·17-s + 1.37·19-s − 0.208·23-s − 25-s + 1.73·27-s + 1.67·29-s + 0.538·31-s − 1.31·37-s + 2.40·39-s − 0.468·41-s + 1.21·43-s + 1.02·47-s + 2.52·51-s − 0.274·53-s + 2.38·57-s + 0.520·59-s + 1.28·61-s − 0.977·67-s − 0.361·69-s − 0.830·71-s − 1.05·73-s − 1.73·75-s + 0.675·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18032\)    =    \(2^{4} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(143.986\)
Root analytic conductor: \(11.9994\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.963508609\)
\(L(\frac12)\) \(\approx\) \(5.963508609\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−15.75491807444898, −15.37059281802214, −14.41704201896826, −14.15449556758356, −13.74916166981985, −13.36603364612210, −12.56702616445881, −12.01214656416754, −11.49114554052111, −10.45142554967340, −10.10097712316345, −9.582662471770889, −8.848518136658713, −8.486431864114987, −7.901326025466960, −7.446914540975553, −6.772736385433112, −5.880766145383648, −5.315433812640729, −4.226505073997403, −3.771153265375360, −3.104003888019642, −2.666555354168430, −1.540026977953522, −1.070853503738618, 1.070853503738618, 1.540026977953522, 2.666555354168430, 3.104003888019642, 3.771153265375360, 4.226505073997403, 5.315433812640729, 5.880766145383648, 6.772736385433112, 7.446914540975553, 7.901326025466960, 8.486431864114987, 8.848518136658713, 9.582662471770889, 10.10097712316345, 10.45142554967340, 11.49114554052111, 12.01214656416754, 12.56702616445881, 13.36603364612210, 13.74916166981985, 14.15449556758356, 14.41704201896826, 15.37059281802214, 15.75491807444898

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