Properties

 Label 2-80e2-1.1-c1-0-54 Degree $2$ Conductor $6400$ Sign $1$ Analytic cond. $51.1042$ Root an. cond. $7.14872$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 3.16·3-s − 4.24·7-s + 7.00·9-s − 13.4·21-s + 1.41·23-s + 12.6·27-s + 8.94·29-s + 12·41-s − 3.16·43-s − 9.89·47-s + 10.9·49-s + 13.4·61-s − 29.6·63-s + 15.8·67-s + 4.47·69-s + 19.0·81-s + 9.48·83-s + 28.2·87-s − 6·89-s − 8.94·101-s + 12.7·103-s + 9.48·107-s + 13.4·109-s + ⋯
 L(s)  = 1 + 1.82·3-s − 1.60·7-s + 2.33·9-s − 2.92·21-s + 0.294·23-s + 2.43·27-s + 1.66·29-s + 1.87·41-s − 0.482·43-s − 1.44·47-s + 1.57·49-s + 1.71·61-s − 3.74·63-s + 1.93·67-s + 0.538·69-s + 2.11·81-s + 1.04·83-s + 3.03·87-s − 0.635·89-s − 0.889·101-s + 1.25·103-s + 0.917·107-s + 1.28·109-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$6400$$    =    $$2^{8} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$51.1042$$ Root analytic conductor: $$7.14872$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{6400} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 6400,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$3.551768507$$ $$L(\frac12)$$ $$\approx$$ $$3.551768507$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 - 3.16T + 3T^{2}$$
7 $$1 + 4.24T + 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 1.41T + 23T^{2}$$
29 $$1 - 8.94T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 - 12T + 41T^{2}$$
43 $$1 + 3.16T + 43T^{2}$$
47 $$1 + 9.89T + 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 13.4T + 61T^{2}$$
67 $$1 - 15.8T + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 9.48T + 83T^{2}$$
89 $$1 + 6T + 89T^{2}$$
97 $$1 + 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

−8.193108199255657759609612910783, −7.38080010297100943189051584426, −6.74912483258790874349853061907, −6.21087955385279688650705187336, −4.97472278276881156017288804561, −4.05169645381293215502582831559, −3.43453838122108151308317996457, −2.83921206954150085600614730334, −2.21372874559264156248051810648, −0.894139317004329303341185546645, 0.894139317004329303341185546645, 2.21372874559264156248051810648, 2.83921206954150085600614730334, 3.43453838122108151308317996457, 4.05169645381293215502582831559, 4.97472278276881156017288804561, 6.21087955385279688650705187336, 6.74912483258790874349853061907, 7.38080010297100943189051584426, 8.193108199255657759609612910783