Properties

Label 2-13-1.1-c3-0-1
Degree $2$
Conductor $13$
Sign $1$
Analytic cond. $0.767024$
Root an. cond. $0.875799$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·2-s − 3.68·3-s − 1.43·4-s + 0.561·5-s − 9.43·6-s + 18.1·7-s − 24.1·8-s − 13.4·9-s + 1.43·10-s + 64.7·11-s + 5.30·12-s − 13·13-s + 46.5·14-s − 2.06·15-s − 50.4·16-s − 25.5·17-s − 34.3·18-s − 107.·19-s − 0.807·20-s − 66.9·21-s + 165.·22-s + 73.2·23-s + 89.0·24-s − 124.·25-s − 33.3·26-s + 148.·27-s − 26.1·28-s + ⋯
L(s)  = 1  + 0.905·2-s − 0.709·3-s − 0.179·4-s + 0.0502·5-s − 0.642·6-s + 0.981·7-s − 1.06·8-s − 0.497·9-s + 0.0454·10-s + 1.77·11-s + 0.127·12-s − 0.277·13-s + 0.888·14-s − 0.0356·15-s − 0.787·16-s − 0.364·17-s − 0.450·18-s − 1.30·19-s − 0.00903·20-s − 0.695·21-s + 1.60·22-s + 0.664·23-s + 0.757·24-s − 0.997·25-s − 0.251·26-s + 1.06·27-s − 0.176·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $1$
Analytic conductor: \(0.767024\)
Root analytic conductor: \(0.875799\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.107518455\)
\(L(\frac12)\) \(\approx\) \(1.107518455\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + 13T \)
good2 \( 1 - 2.56T + 8T^{2} \)
3 \( 1 + 3.68T + 27T^{2} \)
5 \( 1 - 0.561T + 125T^{2} \)
7 \( 1 - 18.1T + 343T^{2} \)
11 \( 1 - 64.7T + 1.33e3T^{2} \)
17 \( 1 + 25.5T + 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 - 73.2T + 1.21e4T^{2} \)
29 \( 1 - 175.T + 2.43e4T^{2} \)
31 \( 1 + 113.T + 2.97e4T^{2} \)
37 \( 1 - 114.T + 5.06e4T^{2} \)
41 \( 1 + 69.6T + 6.89e4T^{2} \)
43 \( 1 - 438.T + 7.95e4T^{2} \)
47 \( 1 + 31.9T + 1.03e5T^{2} \)
53 \( 1 - 2.84T + 1.48e5T^{2} \)
59 \( 1 - 71.6T + 2.05e5T^{2} \)
61 \( 1 + 920.T + 2.26e5T^{2} \)
67 \( 1 + 444.T + 3.00e5T^{2} \)
71 \( 1 + 541.T + 3.57e5T^{2} \)
73 \( 1 - 764.T + 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 - 603.T + 5.71e5T^{2} \)
89 \( 1 + 1.15e3T + 7.04e5T^{2} \)
97 \( 1 - 583.T + 9.12e5T^{2} \)
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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

−19.60374200828868102468536634216, −17.76737342376482072933577699954, −17.04482603597956831823514602140, −14.90239701884330161535029855708, −14.05008445734507485159626055694, −12.28434038437570960343096293189, −11.26038422060788727187354048575, −8.882764193305829362779320448505, −6.17812286510206894290657191844, −4.48460182468764722408570928051, 4.48460182468764722408570928051, 6.17812286510206894290657191844, 8.882764193305829362779320448505, 11.26038422060788727187354048575, 12.28434038437570960343096293189, 14.05008445734507485159626055694, 14.90239701884330161535029855708, 17.04482603597956831823514602140, 17.76737342376482072933577699954, 19.60374200828868102468536634216

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