Properties

Label 2-21-1.1-c29-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.20e4·2-s − 4.78e6·3-s − 3.91e8·4-s − 3.60e9·5-s + 5.76e10·6-s + 6.78e11·7-s + 1.11e13·8-s + 2.28e13·9-s + 4.33e13·10-s + 2.47e15·11-s + 1.87e15·12-s + 1.86e16·13-s − 8.17e15·14-s + 1.72e16·15-s + 7.54e16·16-s − 1.63e17·17-s − 2.75e17·18-s + 4.78e18·19-s + 1.41e18·20-s − 3.24e18·21-s − 2.97e19·22-s + 8.72e19·23-s − 5.35e19·24-s − 1.73e20·25-s − 2.24e20·26-s − 1.09e20·27-s − 2.65e20·28-s + ⋯
L(s)  = 1  − 0.520·2-s − 0.577·3-s − 0.729·4-s − 0.263·5-s + 0.300·6-s + 0.377·7-s + 0.899·8-s + 0.333·9-s + 0.137·10-s + 1.96·11-s + 0.421·12-s + 1.31·13-s − 0.196·14-s + 0.152·15-s + 0.261·16-s − 0.235·17-s − 0.173·18-s + 1.37·19-s + 0.192·20-s − 0.218·21-s − 1.02·22-s + 1.57·23-s − 0.519·24-s − 0.930·25-s − 0.683·26-s − 0.192·27-s − 0.275·28-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: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ 1)\)

Particular Values

\(L(15)\) \(\approx\) \(1.681689473\)
\(L(\frac12)\) \(\approx\) \(1.681689473\)
\(L(\frac{31}{2})\) 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 \( 1 + 1.20e4T + 5.36e8T^{2} \)
5 \( 1 + 3.60e9T + 1.86e20T^{2} \)
11 \( 1 - 2.47e15T + 1.58e30T^{2} \)
13 \( 1 - 1.86e16T + 2.01e32T^{2} \)
17 \( 1 + 1.63e17T + 4.81e35T^{2} \)
19 \( 1 - 4.78e18T + 1.21e37T^{2} \)
23 \( 1 - 8.72e19T + 3.09e39T^{2} \)
29 \( 1 - 4.12e20T + 2.56e42T^{2} \)
31 \( 1 - 2.72e21T + 1.77e43T^{2} \)
37 \( 1 - 8.47e21T + 3.00e45T^{2} \)
41 \( 1 - 1.03e23T + 5.89e46T^{2} \)
43 \( 1 - 9.09e23T + 2.34e47T^{2} \)
47 \( 1 + 1.23e24T + 3.09e48T^{2} \)
53 \( 1 - 9.35e24T + 1.00e50T^{2} \)
59 \( 1 + 4.11e25T + 2.26e51T^{2} \)
61 \( 1 - 6.20e25T + 5.95e51T^{2} \)
67 \( 1 + 3.26e26T + 9.04e52T^{2} \)
71 \( 1 + 1.43e26T + 4.85e53T^{2} \)
73 \( 1 - 7.74e26T + 1.08e54T^{2} \)
79 \( 1 + 4.94e27T + 1.07e55T^{2} \)
83 \( 1 + 6.81e27T + 4.50e55T^{2} \)
89 \( 1 - 2.47e28T + 3.40e56T^{2} \)
97 \( 1 - 8.17e28T + 4.13e57T^{2} \)
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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

−11.86981798797578897812008573897, −11.00303241296693674157700514204, −9.508092753641114212562853659208, −8.689242474880381561366509546368, −7.29300578712274018914198419664, −5.95141323979487359943065647334, −4.54634336422261425702059346578, −3.62925752638482997596555375395, −1.28960731849444808813443017682, −0.866817809499206927815907983602, 0.866817809499206927815907983602, 1.28960731849444808813443017682, 3.62925752638482997596555375395, 4.54634336422261425702059346578, 5.95141323979487359943065647334, 7.29300578712274018914198419664, 8.689242474880381561366509546368, 9.508092753641114212562853659208, 11.00303241296693674157700514204, 11.86981798797578897812008573897

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