Properties

Label 2-29-1.1-c17-0-2
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $53.1344$
Root an. cond. $7.28933$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 39.6·2-s + 1.30e4·3-s − 1.29e5·4-s − 1.72e6·5-s − 5.16e5·6-s − 1.48e7·7-s + 1.03e7·8-s + 4.08e7·9-s + 6.83e7·10-s − 1.36e9·11-s − 1.68e9·12-s − 2.08e9·13-s + 5.87e8·14-s − 2.24e10·15-s + 1.65e10·16-s + 1.13e10·17-s − 1.61e9·18-s − 3.19e10·19-s + 2.23e11·20-s − 1.93e11·21-s + 5.40e10·22-s + 1.33e11·23-s + 1.34e11·24-s + 2.20e12·25-s + 8.27e10·26-s − 1.15e12·27-s + 1.91e12·28-s + ⋯
L(s)  = 1  − 0.109·2-s + 1.14·3-s − 0.988·4-s − 1.97·5-s − 0.125·6-s − 0.971·7-s + 0.217·8-s + 0.316·9-s + 0.216·10-s − 1.91·11-s − 1.13·12-s − 0.709·13-s + 0.106·14-s − 2.26·15-s + 0.964·16-s + 0.394·17-s − 0.0346·18-s − 0.431·19-s + 1.94·20-s − 1.11·21-s + 0.209·22-s + 0.354·23-s + 0.249·24-s + 2.89·25-s + 0.0776·26-s − 0.784·27-s + 0.959·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(53.1344\)
Root analytic conductor: \(7.28933\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(0.1610782603\)
\(L(\frac12)\) \(\approx\) \(0.1610782603\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 5.00e11T \)
good2 \( 1 + 39.6T + 1.31e5T^{2} \)
3 \( 1 - 1.30e4T + 1.29e8T^{2} \)
5 \( 1 + 1.72e6T + 7.62e11T^{2} \)
7 \( 1 + 1.48e7T + 2.32e14T^{2} \)
11 \( 1 + 1.36e9T + 5.05e17T^{2} \)
13 \( 1 + 2.08e9T + 8.65e18T^{2} \)
17 \( 1 - 1.13e10T + 8.27e20T^{2} \)
19 \( 1 + 3.19e10T + 5.48e21T^{2} \)
23 \( 1 - 1.33e11T + 1.41e23T^{2} \)
31 \( 1 + 6.48e12T + 2.25e25T^{2} \)
37 \( 1 + 3.15e12T + 4.56e26T^{2} \)
41 \( 1 + 3.85e13T + 2.61e27T^{2} \)
43 \( 1 - 9.98e13T + 5.87e27T^{2} \)
47 \( 1 + 1.87e14T + 2.66e28T^{2} \)
53 \( 1 + 1.38e14T + 2.05e29T^{2} \)
59 \( 1 + 5.88e14T + 1.27e30T^{2} \)
61 \( 1 + 1.12e15T + 2.24e30T^{2} \)
67 \( 1 + 4.30e15T + 1.10e31T^{2} \)
71 \( 1 + 1.20e15T + 2.96e31T^{2} \)
73 \( 1 - 4.73e15T + 4.74e31T^{2} \)
79 \( 1 - 1.77e16T + 1.81e32T^{2} \)
83 \( 1 - 3.58e16T + 4.21e32T^{2} \)
89 \( 1 + 3.89e16T + 1.37e33T^{2} \)
97 \( 1 - 1.54e16T + 5.95e33T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.18962167674689142088437956892, −12.44045987443904222527794511389, −10.61048826894282167184770740797, −9.204068135287857834483723682703, −8.093725648045483647087227732435, −7.53536874873955172456392078912, −4.94117868856492206394209266387, −3.63235626270815702455873791533, −2.88726995251150686359478211275, −0.20118944854400433629959231473, 0.20118944854400433629959231473, 2.88726995251150686359478211275, 3.63235626270815702455873791533, 4.94117868856492206394209266387, 7.53536874873955172456392078912, 8.093725648045483647087227732435, 9.204068135287857834483723682703, 10.61048826894282167184770740797, 12.44045987443904222527794511389, 13.18962167674689142088437956892

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