Properties

Label 2-13e2-1.1-c9-0-20
Degree $2$
Conductor $169$
Sign $-1$
Analytic cond. $87.0410$
Root an. cond. $9.32957$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.15·2-s − 136.·3-s − 502.·4-s − 2.55e3·5-s + 430.·6-s − 9.39e3·7-s + 3.19e3·8-s − 1.04e3·9-s + 8.04e3·10-s − 4.40e4·11-s + 6.85e4·12-s + 2.96e4·14-s + 3.48e5·15-s + 2.46e5·16-s + 2.82e4·17-s + 3.28e3·18-s − 2.73e5·19-s + 1.28e6·20-s + 1.28e6·21-s + 1.38e5·22-s − 1.12e6·23-s − 4.36e5·24-s + 4.57e6·25-s + 2.82e6·27-s + 4.71e6·28-s − 1.63e6·29-s − 1.09e6·30-s + ⋯
L(s)  = 1  − 0.139·2-s − 0.973·3-s − 0.980·4-s − 1.82·5-s + 0.135·6-s − 1.47·7-s + 0.275·8-s − 0.0529·9-s + 0.254·10-s − 0.908·11-s + 0.954·12-s + 0.206·14-s + 1.77·15-s + 0.942·16-s + 0.0821·17-s + 0.00736·18-s − 0.482·19-s + 1.79·20-s + 1.44·21-s + 0.126·22-s − 0.841·23-s − 0.268·24-s + 2.34·25-s + 1.02·27-s + 1.45·28-s − 0.429·29-s − 0.247·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-1$
Analytic conductor: \(87.0410\)
Root analytic conductor: \(9.32957\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 169,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + 3.15T + 512T^{2} \)
3 \( 1 + 136.T + 1.96e4T^{2} \)
5 \( 1 + 2.55e3T + 1.95e6T^{2} \)
7 \( 1 + 9.39e3T + 4.03e7T^{2} \)
11 \( 1 + 4.40e4T + 2.35e9T^{2} \)
17 \( 1 - 2.82e4T + 1.18e11T^{2} \)
19 \( 1 + 2.73e5T + 3.22e11T^{2} \)
23 \( 1 + 1.12e6T + 1.80e12T^{2} \)
29 \( 1 + 1.63e6T + 1.45e13T^{2} \)
31 \( 1 + 6.65e6T + 2.64e13T^{2} \)
37 \( 1 - 1.71e7T + 1.29e14T^{2} \)
41 \( 1 - 5.15e6T + 3.27e14T^{2} \)
43 \( 1 + 1.97e7T + 5.02e14T^{2} \)
47 \( 1 + 4.82e7T + 1.11e15T^{2} \)
53 \( 1 + 3.06e7T + 3.29e15T^{2} \)
59 \( 1 - 1.15e7T + 8.66e15T^{2} \)
61 \( 1 + 3.62e7T + 1.16e16T^{2} \)
67 \( 1 - 6.48e7T + 2.72e16T^{2} \)
71 \( 1 - 1.47e8T + 4.58e16T^{2} \)
73 \( 1 - 3.37e8T + 5.88e16T^{2} \)
79 \( 1 + 2.04e8T + 1.19e17T^{2} \)
83 \( 1 + 7.61e8T + 1.86e17T^{2} \)
89 \( 1 - 8.29e8T + 3.50e17T^{2} \)
97 \( 1 + 1.00e9T + 7.60e17T^{2} \)
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

−10.70048741746803364187754854417, −9.659475294450197198394423233008, −8.444901642665896602182042141599, −7.59471045212571201007244253730, −6.33454831837771388742661077661, −5.14655760241795873421433752748, −4.05197769984591267103572959706, −3.16808608751511975633311226818, −0.49097619415568430083390199436, 0, 0.49097619415568430083390199436, 3.16808608751511975633311226818, 4.05197769984591267103572959706, 5.14655760241795873421433752748, 6.33454831837771388742661077661, 7.59471045212571201007244253730, 8.444901642665896602182042141599, 9.659475294450197198394423233008, 10.70048741746803364187754854417

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