Properties

Label 2-29-1.1-c9-0-7
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $14.9360$
Root an. cond. $3.86471$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 23.9·2-s + 161.·3-s + 61.9·4-s + 2.64e3·5-s − 3.88e3·6-s + 2.05e3·7-s + 1.07e4·8-s + 6.55e3·9-s − 6.33e4·10-s − 3.15e3·11-s + 1.00e4·12-s − 1.10e5·13-s − 4.93e4·14-s + 4.28e5·15-s − 2.90e5·16-s + 5.71e5·17-s − 1.57e5·18-s − 1.94e5·19-s + 1.63e5·20-s + 3.33e5·21-s + 7.55e4·22-s + 2.29e6·23-s + 1.74e6·24-s + 5.03e6·25-s + 2.64e6·26-s − 2.12e6·27-s + 1.27e5·28-s + ⋯
L(s)  = 1  − 1.05·2-s + 1.15·3-s + 0.120·4-s + 1.89·5-s − 1.22·6-s + 0.324·7-s + 0.930·8-s + 0.333·9-s − 2.00·10-s − 0.0649·11-s + 0.139·12-s − 1.07·13-s − 0.343·14-s + 2.18·15-s − 1.10·16-s + 1.65·17-s − 0.352·18-s − 0.342·19-s + 0.228·20-s + 0.374·21-s + 0.0687·22-s + 1.71·23-s + 1.07·24-s + 2.57·25-s + 1.13·26-s − 0.770·27-s + 0.0392·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(14.9360\)
Root analytic conductor: \(3.86471\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(2.040081574\)
\(L(\frac12)\) \(\approx\) \(2.040081574\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 7.07e5T \)
good2 \( 1 + 23.9T + 512T^{2} \)
3 \( 1 - 161.T + 1.96e4T^{2} \)
5 \( 1 - 2.64e3T + 1.95e6T^{2} \)
7 \( 1 - 2.05e3T + 4.03e7T^{2} \)
11 \( 1 + 3.15e3T + 2.35e9T^{2} \)
13 \( 1 + 1.10e5T + 1.06e10T^{2} \)
17 \( 1 - 5.71e5T + 1.18e11T^{2} \)
19 \( 1 + 1.94e5T + 3.22e11T^{2} \)
23 \( 1 - 2.29e6T + 1.80e12T^{2} \)
31 \( 1 + 2.66e6T + 2.64e13T^{2} \)
37 \( 1 + 9.41e6T + 1.29e14T^{2} \)
41 \( 1 + 3.78e6T + 3.27e14T^{2} \)
43 \( 1 - 3.79e7T + 5.02e14T^{2} \)
47 \( 1 - 1.54e7T + 1.11e15T^{2} \)
53 \( 1 - 6.44e7T + 3.29e15T^{2} \)
59 \( 1 - 4.86e6T + 8.66e15T^{2} \)
61 \( 1 + 7.71e7T + 1.16e16T^{2} \)
67 \( 1 - 3.78e7T + 2.72e16T^{2} \)
71 \( 1 + 1.92e8T + 4.58e16T^{2} \)
73 \( 1 - 6.60e7T + 5.88e16T^{2} \)
79 \( 1 + 2.52e8T + 1.19e17T^{2} \)
83 \( 1 + 2.66e8T + 1.86e17T^{2} \)
89 \( 1 + 6.69e8T + 3.50e17T^{2} \)
97 \( 1 - 8.11e8T + 7.60e17T^{2} \)
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

−14.63642534258086066260330461321, −14.03911368628522366595306933861, −12.91811208040575172320701006024, −10.44983500976842003644628828185, −9.549409836124579618215026153797, −8.810453170826064980555535131289, −7.39067670630845850898336697816, −5.28608430048301561097855406634, −2.62136860136327137453231313146, −1.36710767826383268726192808894, 1.36710767826383268726192808894, 2.62136860136327137453231313146, 5.28608430048301561097855406634, 7.39067670630845850898336697816, 8.810453170826064980555535131289, 9.549409836124579618215026153797, 10.44983500976842003644628828185, 12.91811208040575172320701006024, 14.03911368628522366595306933861, 14.63642534258086066260330461321

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