# Properties

 Label 2-5054-1.1-c1-0-119 Degree $2$ Conductor $5054$ Sign $1$ Analytic cond. $40.3563$ Root an. cond. $6.35266$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 2.59·3-s + 4-s + 3.85·5-s − 2.59·6-s + 7-s − 8-s + 3.70·9-s − 3.85·10-s + 5.26·11-s + 2.59·12-s + 2.42·13-s − 14-s + 9.98·15-s + 16-s − 3.44·17-s − 3.70·18-s + 3.85·20-s + 2.59·21-s − 5.26·22-s − 5.54·23-s − 2.59·24-s + 9.86·25-s − 2.42·26-s + 1.83·27-s + 28-s − 9.13·29-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.49·3-s + 0.5·4-s + 1.72·5-s − 1.05·6-s + 0.377·7-s − 0.353·8-s + 1.23·9-s − 1.21·10-s + 1.58·11-s + 0.747·12-s + 0.673·13-s − 0.267·14-s + 2.57·15-s + 0.250·16-s − 0.834·17-s − 0.874·18-s + 0.861·20-s + 0.565·21-s − 1.12·22-s − 1.15·23-s − 0.528·24-s + 1.97·25-s − 0.476·26-s + 0.353·27-s + 0.188·28-s − 1.69·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5054$$    =    $$2 \cdot 7 \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$40.3563$$ Root analytic conductor: $$6.35266$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{5054} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 5054,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.060198059$$ $$L(\frac12)$$ $$\approx$$ $$4.060198059$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
7 $$1 - T$$
19 $$1$$
good3 $$1 - 2.59T + 3T^{2}$$
5 $$1 - 3.85T + 5T^{2}$$
11 $$1 - 5.26T + 11T^{2}$$
13 $$1 - 2.42T + 13T^{2}$$
17 $$1 + 3.44T + 17T^{2}$$
23 $$1 + 5.54T + 23T^{2}$$
29 $$1 + 9.13T + 29T^{2}$$
31 $$1 + 6.65T + 31T^{2}$$
37 $$1 - 0.133T + 37T^{2}$$
41 $$1 - 6.10T + 41T^{2}$$
43 $$1 - 9.10T + 43T^{2}$$
47 $$1 - 2.28T + 47T^{2}$$
53 $$1 - 5.31T + 53T^{2}$$
59 $$1 - 6.81T + 59T^{2}$$
61 $$1 + 1.44T + 61T^{2}$$
67 $$1 + 5.82T + 67T^{2}$$
71 $$1 + 8.66T + 71T^{2}$$
73 $$1 - 9.83T + 73T^{2}$$
79 $$1 + 0.632T + 79T^{2}$$
83 $$1 + 5.21T + 83T^{2}$$
89 $$1 - 2.58T + 89T^{2}$$
97 $$1 + 1.58T + 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

−8.690261200692748530111565049763, −7.59772800409604809927116940567, −6.99884474384701548919163165888, −6.08371510502231905919670565882, −5.68919604565261845733702902710, −4.21393603350359542965469009276, −3.61941129485597040489987455028, −2.41430379354158142486526244007, −1.95264121133363215172493764349, −1.30601609936167490332960647576, 1.30601609936167490332960647576, 1.95264121133363215172493764349, 2.41430379354158142486526244007, 3.61941129485597040489987455028, 4.21393603350359542965469009276, 5.68919604565261845733702902710, 6.08371510502231905919670565882, 6.99884474384701548919163165888, 7.59772800409604809927116940567, 8.690261200692748530111565049763