# Properties

 Label 2-650-5.4-c5-0-5 Degree $2$ Conductor $650$ Sign $0.447 - 0.894i$ Analytic cond. $104.249$ Root an. cond. $10.2102$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4i·2-s − 16·4-s − 170i·7-s + 64i·8-s + 243·9-s − 250·11-s + 169i·13-s − 680·14-s + 256·16-s + 1.06e3i·17-s − 972i·18-s + 78·19-s + 1.00e3i·22-s − 1.57e3i·23-s + 676·26-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s − 1.31i·7-s + 0.353i·8-s + 9-s − 0.622·11-s + 0.277i·13-s − 0.927·14-s + 0.250·16-s + 0.891i·17-s − 0.707i·18-s + 0.0495·19-s + 0.440i·22-s − 0.621i·23-s + 0.196·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$650$$    =    $$2 \cdot 5^{2} \cdot 13$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$104.249$$ Root analytic conductor: $$10.2102$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{650} (599, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 650,\ (\ :5/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.6100755993$$ $$L(\frac12)$$ $$\approx$$ $$0.6100755993$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 4iT$$
5 $$1$$
13 $$1 - 169iT$$
good3 $$1 - 243T^{2}$$
7 $$1 + 170iT - 1.68e4T^{2}$$
11 $$1 + 250T + 1.61e5T^{2}$$
17 $$1 - 1.06e3iT - 1.41e6T^{2}$$
19 $$1 - 78T + 2.47e6T^{2}$$
23 $$1 + 1.57e3iT - 6.43e6T^{2}$$
29 $$1 + 2.57e3T + 2.05e7T^{2}$$
31 $$1 + 8.65e3T + 2.86e7T^{2}$$
37 $$1 - 1.09e4iT - 6.93e7T^{2}$$
41 $$1 - 1.05e3T + 1.15e8T^{2}$$
43 $$1 - 5.90e3iT - 1.47e8T^{2}$$
47 $$1 + 5.96e3iT - 2.29e8T^{2}$$
53 $$1 + 2.90e4iT - 4.18e8T^{2}$$
59 $$1 - 1.39e4T + 7.14e8T^{2}$$
61 $$1 + 3.28e4T + 8.44e8T^{2}$$
67 $$1 + 6.95e4iT - 1.35e9T^{2}$$
71 $$1 + 5.05e4T + 1.80e9T^{2}$$
73 $$1 - 4.67e4iT - 2.07e9T^{2}$$
79 $$1 - 1.93e4T + 3.07e9T^{2}$$
83 $$1 - 8.74e4iT - 3.93e9T^{2}$$
89 $$1 + 9.41e4T + 5.58e9T^{2}$$
97 $$1 - 1.82e5iT - 8.58e9T^{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

−10.20342167150840282451778053937, −9.327414086266412821706191588042, −8.156622463232810170321130873854, −7.38754431193185287615937658385, −6.49492079448778504743548669428, −5.09010932719553827065645751240, −4.18269143092090988590227267580, −3.48944317235194583476508427491, −1.99381650084970716669695554880, −1.06935338525208003745537233715, 0.13907298106046146538449847430, 1.73768026712978313587296239269, 2.94276659651090590896011864796, 4.24330809710752194713377085436, 5.36825008280670522608614138551, 5.81876445239945086251291291849, 7.19458845854385260263934825074, 7.62842051931812832181152162594, 8.881311486413362419027762561393, 9.338332789163391110963464133790