Properties

Label 2-21-1.1-c29-0-14
Degree $2$
Conductor $21$
Sign $-1$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.24e4·2-s + 4.78e6·3-s − 3.83e8·4-s − 2.55e10·5-s − 5.93e10·6-s + 6.78e11·7-s + 1.14e13·8-s + 2.28e13·9-s + 3.17e14·10-s + 4.13e14·11-s − 1.83e15·12-s − 2.58e16·13-s − 8.41e15·14-s − 1.22e17·15-s + 6.41e16·16-s − 2.80e17·17-s − 2.83e17·18-s + 1.83e18·19-s + 9.79e18·20-s + 3.24e18·21-s − 5.12e18·22-s − 2.39e19·23-s + 5.45e19·24-s + 4.68e20·25-s + 3.20e20·26-s + 1.09e20·27-s − 2.59e20·28-s + ⋯
L(s)  = 1  − 0.535·2-s + 0.577·3-s − 0.713·4-s − 1.87·5-s − 0.309·6-s + 0.377·7-s + 0.917·8-s + 0.333·9-s + 1.00·10-s + 0.328·11-s − 0.411·12-s − 1.81·13-s − 0.202·14-s − 1.08·15-s + 0.222·16-s − 0.403·17-s − 0.178·18-s + 0.526·19-s + 1.33·20-s + 0.218·21-s − 0.175·22-s − 0.430·23-s + 0.529·24-s + 2.51·25-s + 0.973·26-s + 0.192·27-s − 0.269·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-1$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 4.78e6T \)
7 \( 1 - 6.78e11T \)
good2 \( 1 + 1.24e4T + 5.36e8T^{2} \)
5 \( 1 + 2.55e10T + 1.86e20T^{2} \)
11 \( 1 - 4.13e14T + 1.58e30T^{2} \)
13 \( 1 + 2.58e16T + 2.01e32T^{2} \)
17 \( 1 + 2.80e17T + 4.81e35T^{2} \)
19 \( 1 - 1.83e18T + 1.21e37T^{2} \)
23 \( 1 + 2.39e19T + 3.09e39T^{2} \)
29 \( 1 - 1.83e21T + 2.56e42T^{2} \)
31 \( 1 - 2.78e21T + 1.77e43T^{2} \)
37 \( 1 - 6.18e22T + 3.00e45T^{2} \)
41 \( 1 - 3.05e23T + 5.89e46T^{2} \)
43 \( 1 + 3.15e23T + 2.34e47T^{2} \)
47 \( 1 - 1.67e24T + 3.09e48T^{2} \)
53 \( 1 - 2.77e24T + 1.00e50T^{2} \)
59 \( 1 + 4.66e25T + 2.26e51T^{2} \)
61 \( 1 - 9.39e25T + 5.95e51T^{2} \)
67 \( 1 + 3.92e26T + 9.04e52T^{2} \)
71 \( 1 - 7.66e26T + 4.85e53T^{2} \)
73 \( 1 + 1.54e27T + 1.08e54T^{2} \)
79 \( 1 + 5.44e27T + 1.07e55T^{2} \)
83 \( 1 + 6.85e27T + 4.50e55T^{2} \)
89 \( 1 - 8.95e27T + 3.40e56T^{2} \)
97 \( 1 - 5.94e28T + 4.13e57T^{2} \)
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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

−11.60223162413638545655944403832, −10.10569288192407514230945649238, −8.864938158275430298494210222878, −7.900502469558541191000780305159, −7.28608460700529218800337738656, −4.71875223320364337008037720763, −4.15055843571398938234569588143, −2.75285510768904842228894166992, −0.956972794895761116246775721953, 0, 0.956972794895761116246775721953, 2.75285510768904842228894166992, 4.15055843571398938234569588143, 4.71875223320364337008037720763, 7.28608460700529218800337738656, 7.900502469558541191000780305159, 8.864938158275430298494210222878, 10.10569288192407514230945649238, 11.60223162413638545655944403832

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