# Properties

 Label 2-338-1.1-c3-0-28 Degree $2$ Conductor $338$ Sign $-1$ Analytic cond. $19.9426$ Root an. cond. $4.46571$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·2-s + 0.405·3-s + 4·4-s + 6.36·5-s − 0.811·6-s − 2.55·7-s − 8·8-s − 26.8·9-s − 12.7·10-s + 26.1·11-s + 1.62·12-s + 5.10·14-s + 2.58·15-s + 16·16-s − 93.7·17-s + 53.6·18-s − 37.2·19-s + 25.4·20-s − 1.03·21-s − 52.2·22-s − 104.·23-s − 3.24·24-s − 84.5·25-s − 21.8·27-s − 10.2·28-s + 249.·29-s − 5.16·30-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.0780·3-s + 0.5·4-s + 0.569·5-s − 0.0552·6-s − 0.137·7-s − 0.353·8-s − 0.993·9-s − 0.402·10-s + 0.715·11-s + 0.0390·12-s + 0.0973·14-s + 0.0444·15-s + 0.250·16-s − 1.33·17-s + 0.702·18-s − 0.449·19-s + 0.284·20-s − 0.0107·21-s − 0.506·22-s − 0.951·23-s − 0.0276·24-s − 0.676·25-s − 0.155·27-s − 0.0688·28-s + 1.59·29-s − 0.0314·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$338$$    =    $$2 \cdot 13^{2}$$ Sign: $-1$ Analytic conductor: $$19.9426$$ Root analytic conductor: $$4.46571$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 338,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1 + 2T$$
13 $$1$$
good3 $$1 - 0.405T + 27T^{2}$$
5 $$1 - 6.36T + 125T^{2}$$
7 $$1 + 2.55T + 343T^{2}$$
11 $$1 - 26.1T + 1.33e3T^{2}$$
17 $$1 + 93.7T + 4.91e3T^{2}$$
19 $$1 + 37.2T + 6.85e3T^{2}$$
23 $$1 + 104.T + 1.21e4T^{2}$$
29 $$1 - 249.T + 2.43e4T^{2}$$
31 $$1 - 278.T + 2.97e4T^{2}$$
37 $$1 + 10.9T + 5.06e4T^{2}$$
41 $$1 + 371.T + 6.89e4T^{2}$$
43 $$1 + 413.T + 7.95e4T^{2}$$
47 $$1 - 238.T + 1.03e5T^{2}$$
53 $$1 + 424.T + 1.48e5T^{2}$$
59 $$1 + 774.T + 2.05e5T^{2}$$
61 $$1 + 123.T + 2.26e5T^{2}$$
67 $$1 + 881.T + 3.00e5T^{2}$$
71 $$1 - 118.T + 3.57e5T^{2}$$
73 $$1 - 209.T + 3.89e5T^{2}$$
79 $$1 + 532.T + 4.93e5T^{2}$$
83 $$1 + 376.T + 5.71e5T^{2}$$
89 $$1 - 42.6T + 7.04e5T^{2}$$
97 $$1 - 639.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

−10.48507400452114034455344179041, −9.688010731060488302240715097642, −8.742054684412510920450851833998, −8.139381257562754021363752272394, −6.58389709471778843864988520433, −6.14345367119653406161522408975, −4.59548535936550969534478016969, −2.99076026862635851518010660339, −1.78465533015960456163356550174, 0, 1.78465533015960456163356550174, 2.99076026862635851518010660339, 4.59548535936550969534478016969, 6.14345367119653406161522408975, 6.58389709471778843864988520433, 8.139381257562754021363752272394, 8.742054684412510920450851833998, 9.688010731060488302240715097642, 10.48507400452114034455344179041