# Properties

 Label 2-13e2-1.1-c1-0-3 Degree $2$ Conductor $169$ Sign $-1$ Analytic cond. $1.34947$ Root an. cond. $1.16166$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.24·2-s − 0.554·3-s + 3.04·4-s − 1.44·5-s + 1.24·6-s + 2.04·7-s − 2.35·8-s − 2.69·9-s + 3.24·10-s − 2.55·11-s − 1.69·12-s − 4.60·14-s + 0.801·15-s − 0.801·16-s − 5.29·17-s + 6.04·18-s − 5.85·19-s − 4.40·20-s − 1.13·21-s + 5.74·22-s − 1.89·23-s + 1.30·24-s − 2.91·25-s + 3.15·27-s + 6.24·28-s + 2.26·29-s − 1.80·30-s + ⋯
 L(s)  = 1 − 1.58·2-s − 0.320·3-s + 1.52·4-s − 0.646·5-s + 0.509·6-s + 0.774·7-s − 0.833·8-s − 0.897·9-s + 1.02·10-s − 0.770·11-s − 0.488·12-s − 1.23·14-s + 0.207·15-s − 0.200·16-s − 1.28·17-s + 1.42·18-s − 1.34·19-s − 0.985·20-s − 0.248·21-s + 1.22·22-s − 0.394·23-s + 0.266·24-s − 0.582·25-s + 0.607·27-s + 1.18·28-s + 0.421·29-s − 0.328·30-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$169$$    =    $$13^{2}$$ Sign: $-1$ Analytic conductor: $$1.34947$$ Root analytic conductor: $$1.16166$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 169,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 + 2.24T + 2T^{2}$$
3 $$1 + 0.554T + 3T^{2}$$
5 $$1 + 1.44T + 5T^{2}$$
7 $$1 - 2.04T + 7T^{2}$$
11 $$1 + 2.55T + 11T^{2}$$
17 $$1 + 5.29T + 17T^{2}$$
19 $$1 + 5.85T + 19T^{2}$$
23 $$1 + 1.89T + 23T^{2}$$
29 $$1 - 2.26T + 29T^{2}$$
31 $$1 + 4.26T + 31T^{2}$$
37 $$1 - 5.35T + 37T^{2}$$
41 $$1 - 1.27T + 41T^{2}$$
43 $$1 - 6.13T + 43T^{2}$$
47 $$1 + 2.95T + 47T^{2}$$
53 $$1 - 5.52T + 53T^{2}$$
59 $$1 + 12.2T + 59T^{2}$$
61 $$1 - 8.56T + 61T^{2}$$
67 $$1 - 0.576T + 67T^{2}$$
71 $$1 + 4.59T + 71T^{2}$$
73 $$1 + 10.5T + 73T^{2}$$
79 $$1 + 15.7T + 79T^{2}$$
83 $$1 - 7.72T + 83T^{2}$$
89 $$1 - 6.61T + 89T^{2}$$
97 $$1 - 11.9T + 97T^{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

−11.68876898935071458998037617008, −11.09985480453711155356091186659, −10.40712211024041910491450461924, −8.973815934188217580884209870242, −8.296094328891065865091021257589, −7.51191873285717262157202617450, −6.16677339140566145731511886704, −4.52958182550039471163474635583, −2.27315991722013939124163957551, 0, 2.27315991722013939124163957551, 4.52958182550039471163474635583, 6.16677339140566145731511886704, 7.51191873285717262157202617450, 8.296094328891065865091021257589, 8.973815934188217580884209870242, 10.40712211024041910491450461924, 11.09985480453711155356091186659, 11.68876898935071458998037617008