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Properties

Label 2-20339-1.1-c1-0-4
Degree $2$
Conductor $20339$
Sign $1$
Analytic cond. $162.407$
Root an. cond. $12.7439$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 3·5-s − 6-s + 7-s + 3·8-s − 2·9-s + 3·10-s − 11-s − 12-s − 3·13-s − 14-s − 3·15-s − 16-s − 3·17-s + 2·18-s − 19-s + 3·20-s + 21-s + 22-s + 7·23-s + 3·24-s + 4·25-s + 3·26-s − 5·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.774·15-s − 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s + 0.670·20-s + 0.218·21-s + 0.213·22-s + 1.45·23-s + 0.612·24-s + 4/5·25-s + 0.588·26-s − 0.962·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20339\)    =    \(11 \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(162.407\)
Root analytic conductor: \(12.7439\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 20339,\ (\ :1/2),\ 1)\)

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)$
bad11 \( 1 + T \)
43 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
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

−16.34120108213021, −15.43897936481234, −15.14825689319415, −14.62031304236794, −14.19855674772293, −13.35739386212576, −13.01152070096201, −12.39524248927956, −11.51355808863007, −11.26527158176566, −10.72275304109383, −9.978734493665595, −9.254103573747507, −8.807933654293551, −8.470598723114904, −7.778375586637309, −7.382003282051828, −6.962936556369720, −5.688908428142141, −5.059879729415071, −4.492853881829228, −3.796568625147948, −3.208372315263622, −2.325380136163658, −1.441706440749290, 0, 0, 1.441706440749290, 2.325380136163658, 3.208372315263622, 3.796568625147948, 4.492853881829228, 5.059879729415071, 5.688908428142141, 6.962936556369720, 7.382003282051828, 7.778375586637309, 8.470598723114904, 8.807933654293551, 9.254103573747507, 9.978734493665595, 10.72275304109383, 11.26527158176566, 11.51355808863007, 12.39524248927956, 13.01152070096201, 13.35739386212576, 14.19855674772293, 14.62031304236794, 15.14825689319415, 15.43897936481234, 16.34120108213021

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