Properties

Label 2-18032-1.1-c1-0-5
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  + 2·3-s + 9-s − 4·11-s − 6·17-s − 6·19-s + 23-s − 5·25-s − 4·27-s + 10·29-s + 4·31-s − 8·33-s − 2·37-s + 10·41-s + 4·43-s + 12·47-s − 12·51-s − 6·53-s − 12·57-s − 2·59-s + 2·69-s + 8·71-s + 6·73-s − 10·75-s + 8·79-s − 11·81-s − 14·83-s + 20·87-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.20·11-s − 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 0.769·27-s + 1.85·29-s + 0.718·31-s − 1.39·33-s − 0.328·37-s + 1.56·41-s + 0.609·43-s + 1.75·47-s − 1.68·51-s − 0.824·53-s − 1.58·57-s − 0.260·59-s + 0.240·69-s + 0.949·71-s + 0.702·73-s − 1.15·75-s + 0.900·79-s − 1.22·81-s − 1.53·83-s + 2.14·87-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: $\chi_{18032} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207647151\)
\(L(\frac12)\) \(\approx\) \(2.207647151\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.62989906641987, −15.41865232385575, −14.66071731993211, −14.10177589376803, −13.63285645348876, −13.22882516987438, −12.61640426312756, −12.09312721374446, −11.18211594307814, −10.70226450822257, −10.27677414962586, −9.378960149914431, −9.037845264286810, −8.228950472825199, −8.130410484319855, −7.361493493800661, −6.572385053903040, −6.045862414651080, −5.178994829561110, −4.359991558894608, −4.008936317936301, −2.857043682222406, −2.532055148735787, −1.955969824435345, −0.5487632185387970, 0.5487632185387970, 1.955969824435345, 2.532055148735787, 2.857043682222406, 4.008936317936301, 4.359991558894608, 5.178994829561110, 6.045862414651080, 6.572385053903040, 7.361493493800661, 8.130410484319855, 8.228950472825199, 9.037845264286810, 9.378960149914431, 10.27677414962586, 10.70226450822257, 11.18211594307814, 12.09312721374446, 12.61640426312756, 13.22882516987438, 13.63285645348876, 14.10177589376803, 14.66071731993211, 15.41865232385575, 15.62989906641987

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