Properties

Label 2-13-1.1-c5-0-2
Degree $2$
Conductor $13$
Sign $1$
Analytic cond. $2.08498$
Root an. cond. $1.44394$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 10.9·2-s − 15.7·3-s + 88.2·4-s − 51.6·5-s − 173.·6-s − 75.3·7-s + 617.·8-s + 6.60·9-s − 565.·10-s + 255.·11-s − 1.39e3·12-s + 169·13-s − 826.·14-s + 815.·15-s + 3.94e3·16-s + 53.2·17-s + 72.4·18-s + 268.·19-s − 4.55e3·20-s + 1.19e3·21-s + 2.80e3·22-s + 2.08e3·23-s − 9.75e3·24-s − 461.·25-s + 1.85e3·26-s + 3.73e3·27-s − 6.65e3·28-s + ⋯
L(s)  = 1  + 1.93·2-s − 1.01·3-s + 2.75·4-s − 0.923·5-s − 1.96·6-s − 0.581·7-s + 3.41·8-s + 0.0271·9-s − 1.78·10-s + 0.637·11-s − 2.79·12-s + 0.277·13-s − 1.12·14-s + 0.935·15-s + 3.85·16-s + 0.0446·17-s + 0.0527·18-s + 0.170·19-s − 2.54·20-s + 0.589·21-s + 1.23·22-s + 0.820·23-s − 3.45·24-s − 0.147·25-s + 0.537·26-s + 0.985·27-s − 1.60·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $1$
Analytic conductor: \(2.08498\)
Root analytic conductor: \(1.44394\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.328309214\)
\(L(\frac12)\) \(\approx\) \(2.328309214\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - 169T \)
good2 \( 1 - 10.9T + 32T^{2} \)
3 \( 1 + 15.7T + 243T^{2} \)
5 \( 1 + 51.6T + 3.12e3T^{2} \)
7 \( 1 + 75.3T + 1.68e4T^{2} \)
11 \( 1 - 255.T + 1.61e5T^{2} \)
17 \( 1 - 53.2T + 1.41e6T^{2} \)
19 \( 1 - 268.T + 2.47e6T^{2} \)
23 \( 1 - 2.08e3T + 6.43e6T^{2} \)
29 \( 1 + 8.17e3T + 2.05e7T^{2} \)
31 \( 1 + 4.78e3T + 2.86e7T^{2} \)
37 \( 1 + 6.65e3T + 6.93e7T^{2} \)
41 \( 1 - 1.28e3T + 1.15e8T^{2} \)
43 \( 1 - 3.80e3T + 1.47e8T^{2} \)
47 \( 1 - 3.77e3T + 2.29e8T^{2} \)
53 \( 1 - 2.40e4T + 4.18e8T^{2} \)
59 \( 1 - 2.45e4T + 7.14e8T^{2} \)
61 \( 1 - 3.51e4T + 8.44e8T^{2} \)
67 \( 1 - 1.50e4T + 1.35e9T^{2} \)
71 \( 1 - 5.11e4T + 1.80e9T^{2} \)
73 \( 1 + 6.30e4T + 2.07e9T^{2} \)
79 \( 1 + 8.04e4T + 3.07e9T^{2} \)
83 \( 1 - 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + 6.95e4T + 5.58e9T^{2} \)
97 \( 1 - 1.83e4T + 8.58e9T^{2} \)
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

−19.36350207000099650931787427913, −16.82314098688798847438679660234, −15.88959419808656649104500497291, −14.65382503537125910081734629899, −13.00977274117946364462952881084, −11.86400938937948299853532355758, −11.06304313393572815208778688582, −6.98817305070137726126407363662, −5.54746412877012370305582877523, −3.74784729016099998840520889222, 3.74784729016099998840520889222, 5.54746412877012370305582877523, 6.98817305070137726126407363662, 11.06304313393572815208778688582, 11.86400938937948299853532355758, 13.00977274117946364462952881084, 14.65382503537125910081734629899, 15.88959419808656649104500497291, 16.82314098688798847438679660234, 19.36350207000099650931787427913

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