Properties

Label 2-29-1.1-c5-0-9
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.16·2-s − 17.5·3-s − 14.6·4-s + 32.5·5-s − 72.9·6-s − 220.·7-s − 194.·8-s + 64.1·9-s + 135.·10-s − 85.7·11-s + 257.·12-s + 1.03e3·13-s − 919.·14-s − 570.·15-s − 339.·16-s − 313.·17-s + 267.·18-s + 458.·19-s − 477.·20-s + 3.86e3·21-s − 356.·22-s − 3.44e3·23-s + 3.40e3·24-s − 2.06e3·25-s + 4.30e3·26-s + 3.13e3·27-s + 3.24e3·28-s + ⋯
L(s)  = 1  + 0.735·2-s − 1.12·3-s − 0.458·4-s + 0.582·5-s − 0.827·6-s − 1.70·7-s − 1.07·8-s + 0.264·9-s + 0.428·10-s − 0.213·11-s + 0.515·12-s + 1.69·13-s − 1.25·14-s − 0.654·15-s − 0.331·16-s − 0.262·17-s + 0.194·18-s + 0.291·19-s − 0.267·20-s + 1.91·21-s − 0.157·22-s − 1.35·23-s + 1.20·24-s − 0.660·25-s + 1.24·26-s + 0.827·27-s + 0.781·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + 841T \)
good2 \( 1 - 4.16T + 32T^{2} \)
3 \( 1 + 17.5T + 243T^{2} \)
5 \( 1 - 32.5T + 3.12e3T^{2} \)
7 \( 1 + 220.T + 1.68e4T^{2} \)
11 \( 1 + 85.7T + 1.61e5T^{2} \)
13 \( 1 - 1.03e3T + 3.71e5T^{2} \)
17 \( 1 + 313.T + 1.41e6T^{2} \)
19 \( 1 - 458.T + 2.47e6T^{2} \)
23 \( 1 + 3.44e3T + 6.43e6T^{2} \)
31 \( 1 + 7.98e3T + 2.86e7T^{2} \)
37 \( 1 - 152.T + 6.93e7T^{2} \)
41 \( 1 + 1.84e4T + 1.15e8T^{2} \)
43 \( 1 - 2.07e3T + 1.47e8T^{2} \)
47 \( 1 - 1.58e4T + 2.29e8T^{2} \)
53 \( 1 - 9.24e3T + 4.18e8T^{2} \)
59 \( 1 + 1.43e4T + 7.14e8T^{2} \)
61 \( 1 + 1.95e4T + 8.44e8T^{2} \)
67 \( 1 + 9.19e3T + 1.35e9T^{2} \)
71 \( 1 + 1.93e4T + 1.80e9T^{2} \)
73 \( 1 + 5.69e4T + 2.07e9T^{2} \)
79 \( 1 - 5.15e4T + 3.07e9T^{2} \)
83 \( 1 - 1.99e4T + 3.93e9T^{2} \)
89 \( 1 - 1.30e5T + 5.58e9T^{2} \)
97 \( 1 - 4.36e4T + 8.58e9T^{2} \)
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

−15.66891952819415702992310901843, −13.77335358258133985438236697668, −13.08940155189162677732128584854, −11.95045393567753199675588338820, −10.37312581913369923034260356122, −9.072611988000810662112591225342, −6.28412900245602665615722872437, −5.70081226594538631404387762560, −3.63547011652282956089496635579, 0, 3.63547011652282956089496635579, 5.70081226594538631404387762560, 6.28412900245602665615722872437, 9.072611988000810662112591225342, 10.37312581913369923034260356122, 11.95045393567753199675588338820, 13.08940155189162677732128584854, 13.77335358258133985438236697668, 15.66891952819415702992310901843

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