Properties

Label 2-13e2-1.1-c3-0-27
Degree $2$
Conductor $169$
Sign $-1$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.46·2-s − 7·3-s + 3.99·4-s + 13.8·5-s − 24.2·6-s − 22.5·7-s − 13.8·8-s + 22·9-s + 47.9·10-s − 22.5·11-s − 27.9·12-s − 77.9·14-s − 96.9·15-s − 80·16-s − 27·17-s + 76.2·18-s − 88.3·19-s + 55.4·20-s + 157.·21-s − 77.9·22-s − 57·23-s + 96.9·24-s + 66.9·25-s + 35·27-s − 90.0·28-s − 69·29-s − 336·30-s + ⋯
L(s)  = 1  + 1.22·2-s − 1.34·3-s + 0.499·4-s + 1.23·5-s − 1.64·6-s − 1.21·7-s − 0.612·8-s + 0.814·9-s + 1.51·10-s − 0.617·11-s − 0.673·12-s − 1.48·14-s − 1.66·15-s − 1.25·16-s − 0.385·17-s + 0.997·18-s − 1.06·19-s + 0.619·20-s + 1.63·21-s − 0.755·22-s − 0.516·23-s + 0.824·24-s + 0.535·25-s + 0.249·27-s − 0.607·28-s − 0.441·29-s − 2.04·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-1$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 - 3.46T + 8T^{2} \)
3 \( 1 + 7T + 27T^{2} \)
5 \( 1 - 13.8T + 125T^{2} \)
7 \( 1 + 22.5T + 343T^{2} \)
11 \( 1 + 22.5T + 1.33e3T^{2} \)
17 \( 1 + 27T + 4.91e3T^{2} \)
19 \( 1 + 88.3T + 6.85e3T^{2} \)
23 \( 1 + 57T + 1.21e4T^{2} \)
29 \( 1 + 69T + 2.43e4T^{2} \)
31 \( 1 + 72.7T + 2.97e4T^{2} \)
37 \( 1 + 39.8T + 5.06e4T^{2} \)
41 \( 1 - 393.T + 6.89e4T^{2} \)
43 \( 1 - 85T + 7.95e4T^{2} \)
47 \( 1 - 342.T + 1.03e5T^{2} \)
53 \( 1 - 426T + 1.48e5T^{2} \)
59 \( 1 - 19.0T + 2.05e5T^{2} \)
61 \( 1 + 17T + 2.26e5T^{2} \)
67 \( 1 + 164.T + 3.00e5T^{2} \)
71 \( 1 - 583.T + 3.57e5T^{2} \)
73 \( 1 + 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 + 426.T + 5.71e5T^{2} \)
89 \( 1 - 306.T + 7.04e5T^{2} \)
97 \( 1 - 1.23e3T + 9.12e5T^{2} \)
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

−12.19897748690645773708945667614, −10.96516330966958945477635777170, −10.08718045321594415059779411528, −9.057830829693235212687455397755, −6.81711797381145033805839784079, −5.94522954094110184298884003891, −5.58197403699315781284562515063, −4.23025508461146700014839836048, −2.54707575418626506781257122783, 0, 2.54707575418626506781257122783, 4.23025508461146700014839836048, 5.58197403699315781284562515063, 5.94522954094110184298884003891, 6.81711797381145033805839784079, 9.057830829693235212687455397755, 10.08718045321594415059779411528, 10.96516330966958945477635777170, 12.19897748690645773708945667614

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