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Properties

Label 2-1729-1729.909-c0-0-1
Degree $2$
Conductor $1729$
Sign $0.188 + 0.982i$
Analytic cond. $0.862883$
Root an. cond. $0.928915$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)4-s + (0.266 − 1.50i)5-s + (−0.5 − 0.866i)7-s + (0.766 − 0.642i)9-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)16-s + (−0.939 − 0.342i)19-s + 1.53·20-s + (−0.326 − 1.85i)23-s + (−1.26 − 0.460i)25-s + (0.766 − 0.642i)28-s + (0.266 − 0.223i)29-s + (0.939 + 1.62i)31-s + (−1.43 + 0.524i)35-s + (0.766 + 0.642i)36-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)4-s + (0.266 − 1.50i)5-s + (−0.5 − 0.866i)7-s + (0.766 − 0.642i)9-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)16-s + (−0.939 − 0.342i)19-s + 1.53·20-s + (−0.326 − 1.85i)23-s + (−1.26 − 0.460i)25-s + (0.766 − 0.642i)28-s + (0.266 − 0.223i)29-s + (0.939 + 1.62i)31-s + (−1.43 + 0.524i)35-s + (0.766 + 0.642i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1729\)    =    \(7 \cdot 13 \cdot 19\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(0.862883\)
Root analytic conductor: \(0.928915\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1729} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1729,\ (\ :0),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056460484\)
\(L(\frac12)\) \(\approx\) \(1.056460484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
good2 \( 1 + (-0.173 - 0.984i)T^{2} \)
3 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.244535206112322831080608031771, −8.501012942499737666694970172218, −7.930337381252079850170312912167, −6.84162317345569643016803463894, −6.45389324200249666485865477442, −4.81584979467809288147122475868, −4.49945609469761030699798595090, −3.53460767610971748990500237097, −2.30625067892398241866938885094, −0.78582813699862297376178584128, 2.02222328461632764615543637239, 2.42089271451605913738048183994, 3.72652202419747591867630060509, 4.95316594160874258564075468385, 5.81002164108977460133807164340, 6.47458892560535199692076152432, 7.10712346181281569380014080367, 7.910215260092651055503069498361, 9.257871022721793123547501567167, 9.897918003390585446230958313186

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